MODULI SPACES OF ABELIAN DIFFERENTIALS: THE PRINCIPAL BOUNDARY, COUNTING PROBLEMS

AND THE SIEGEL–VEECH CONSTANTS

ALEX ESKIN, HOWARD MASUR, AND ANTON ZORICH

Abstract. A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment (saddle connection) joining a pair of conical points one can find with a nonzero probability another saddle connection on S having the same direction and the same length as the initial one. The similar phenomenon is valid for the families of parallel closed geodesics.

We give a complete description of all possible configurations of parallel saddle connections (and of families of parallel closed geodesics) which might be found on a generic flat surface S. We count the number of saddle connections of length less than L on a generic flat surface S; we also count the number of admissible configurations of pairs (triples,...) of saddle connections; we count the analogous numbers of configurations of families of closed geodesics. By the previous result of [EMa] these numbers have quadratic asymptotics c · (πL^{2}). Here we explicitly compute the constant c for a configuration of every type. The constant c is found from a Siegel–Veech formula.

To perform this computation we elaborate the detailed description of the principal part of the boundary of the moduli space of holomorphic 1-forms and we find the numerical value of the normalized volume of the tubular neighborhood of the boundary. We use this for evaluation of integrals over the moduli space.

Contents

Introduction 2 1. Flat Surfaces and Geodesics on Flat Surfaces 3 1.1. Flat Surfaces and Abelian Differentials 5 1.2. Holonomy and the sets V_{cyl}(S) and V_{sc}(S). 6 2. Moduli Spaces of Abelian Differentials 6 2.1. Stratification 6 2.2. Local Coordinates and Volume Element 7 3. Evaluation of the Constants 9 3.1. Configurations of Saddle Connections 9 3.2. Siegel–Veech Formula and Computation of the Constant 11 4. Readers Guide 13 5. Connected Components of the Strata 13 5.1. Parity of a Spin Structure 13 5.2. Connected Components of the Strata 14 5.3. Strata of Surfaces with Marked Points 15

Research of the first author is partially supported by NSF grant DMS-9704845, the Sloan Foundation and the Packard Foundation.

Research of the second author is partially supported by NSF grant 9803497.

Part 1. Saddle Connections Joining Distinct Zeroes 15 6. Saddle Connections of Multiplicity One. Connected Strata 16 6.1. Collapsing a Pair of Zeroes, Principal Boundary 16 6.2. Breaking up a Zero 16 6.3. Computing the Asymptotic Constants 18 6.4. Examples: Constants for Connected Strata in Genus 3 20 6.5. Constants for the Principal Stratum H_{1}(1, …, 1) 22 7. Multiple Homologous Saddle Connections. Connected Strata 22 7.1. Principal Boundary 22 7.2. Slit Construction 23 7.3. Building Surfaces With Multiple Saddle Connections 23 7.4. Stratum Interchange and γ → −γ Symmetry 25 7.5. The Constant in Higher Multiplicity, Connected Strata 28 7.6. Principal stratum 31 8. Strata that are not Connected 32 8.1. Parity of the Spin Structure in the Slit Construction. Admissible

Assignments for Surfaces from Hyperelliptic Components 32

8.2. Constants for the Components of the Stratum H_{1}(g − 1, g − 1) 33 8.3. Other Strata 39

Part 2. Saddle Connections Joining a Zero to Itself 41 9. Principal Boundary 41 10. Constructing surfaces with homologous saddle connections 44 10.1. Figure Eight Construction 44 10.2. Parallelogram Construction 45 10.3. Admissible Constructions 46 10.4. Stratum Interchange and γ → −γ Symmetry 49 10.5. Combinatorial Factor, Connected Strata 52 11. Computing the Constants for Connected Strata 54 11.1. Computation of the constants 54 11.2. Examples: Constants for Strata in Genus 3 57 11.3. Principal stratum 59 12. Strata that are not Connected 62 12.1. Parity of the Spin Structure of the Compound Surface 62 12.2. Decomposition of Surfaces from Hyperelliptic Components 65 12.3. Constants for the Hyperelliptic Connected Components 66 12.4. Connected Component H^{nonhyp}(g − 1, g − 1); Even Genus g 69 12.5. Nonhyperelliptic Components with a Spin Structure 70

Appendix. Values of the Constants 75 Appendix A. Saddle Connections Joining Distinct Zeroes 75 A.1. Connected Strata in Genus 4 75 Appendix B. Saddle Connections Joining a Zero to Itself 79 B.1. Connected Strata in Genus 4 79 B.2. Closed Geodesics for Nonconnected Strata in Genus 4 85 References 88

Introduction

1. Flat Surfaces and Geodesics on Flat Surfaces

In this paper we shall consider flat metrics with isolated conical singularities on a closed orientable surface of genus g. An important class of such flat metrics corresponds to the translation surfaces: those surfaces for which the flat metric on the surface has trivial linear holonomy. These surfaces have been studied by several authors in various guises; in the context of Abelian differentials (and more generally quadratic differentials) on compact Riemann surface in [S], in the context of rational billiards in [KaZe], [KMS], [Ve3], and [GJ]. Moduli spaces of these structures have been studied in [Ve4 ], [MS], [Ko], [KonZo], and [EMa]. A survey is given in [MT].

Now the trivial holonomy means, in particular, that the parallel transport of a vector along a small loop going around a conical point brings a vector back to itself. This implies that all cone angles of such surface are integer multiples of 2π.

Chose a direction in the tangent space to some base point x_{0} ∈ S of the flat surface and transport this direction using the parallel transport to all nonsingular points of the surface. Since the monodromy is trivial this parallel transport does not depend on the path. Thus any direction is globally defined on the flat surface punctured at the singularities. Throughout this paper by a flat surface we mean a flat surface having trivial holonomy representation in SO(2, ℝ) endowed with a choice of direction. Following a tradition we often call the conical points the saddles.

Figure 1. Identifying the opposite sides of a regular octagon by parallel translations we obtain a flat surface of genus g = 2. Deforming the regular octagon in such way that the sides are organized into pairs of parallel sides of equal length we get a family of flat surfaces of genus g = 2 each having a single conical singularity with the cone angle 6π.

A flat torus endowed with a choice of direction gives an example of a flat surface without any saddles. One can glue a flat torus from a parallelogram by identifying opposite sides. A generalization can be found by considering a polygon in the plane having the property that its sides are distributed into pairs that are parallel and of equal length. Identifying the pairs by translations one gets a closed oriented flat surface. For example, identifying the opposite sides of a regular octagon one gets a surface of genus 2, see Figure 1. All vertices of the octagon are identified to a single saddle which has an angle of 6π. We may assume that we have cartesian coordinates in the polygon; by convention we chose the vertical direction as the distinguished one.

In these coordinates away from the saddles, geodesics on the surface S are straight lines. The behavior of geodesics on flat surfaces is in many aspects similar to the behavior of geodesics on a flat torus. In particular, a geodesic cannot change the direction, and thus cannot have self intersections. Generically a geodesic is dense in the surface which is analogous to an irrational winding line on the torus.

However, a flat surface different from the torus must have saddles; some geodesics hit the saddles. Some geodesics hit the saddles going both in forward and in backward directions. Such a geodesic segment joining a pair of saddles and having no saddles in its interior is called a saddle connection.

A geodesic leaving a regular point x may return back to x without meeting any singularities. Since it cannot change its direction, we necessarily get a closed regular geodesic in this case. Note that as in the case of the torus, any geodesic leaving a nearby point in a parallel direction will be also closed and will have the same length as the initial one. Thus closed regular geodesics appear in families of parallel geodesics of the same length. However, unlike the case of the torus, such parallel closed geodesics do not typically fill all the surface, but only a cylindrical subset. Each boundary component of such a cylinder contains a saddle. For example, choosing the vertical direction on the flat surface obtained from a regular octagon (see Figure 1) we get two families of regular closed geodesics, and each of the two families is filling a cylinder.

Note that the shortest representative in a homotopy class of curves joining two saddles exists. Typically, even for the class of a simple closed curve, one gets a broken line containing many geodesic segments. The segments of these broken lines pass through the saddles changing directions there. We want to stress that throughout this paper we consider only those saddle connections which contain a single geodesic segment and thus no saddles in its interior. However, we do not exclude the situation when the endpoints of a saddle connection coincide. We call such saddle connections the closed geodesics.

Now fix a flat surface S. Let Ŝ be the universal cover of S − Σ, where Σ is the collection of saddles. There is an isometric map of Ŝ to ℝ^{2} called the developing map which maps a lift γ̂ of an oriented curve γ on S to a curve in ℝ^{2}. We denote by hol(γ) the difference of the endpoints of the image. The vector hol(γ) ∈ ℝ^{2} is called the holonomy vector of γ.

Remark 1.1. By definition a flat surface has trivial holonomy representation in the linear group

π_{1}(S) → H_{1}(S, ℤ) → SO(2, ℝ).

However, the holonomy representation in the affine group is nontrivial: its image belongs to the group of translations. From now on we reserve the notion of holonomy for this second representation.

Now fix a pair of saddles on a flat surface S, and some length L. Consider all those saddle connections joining the fixed saddles whose length is shorter then L. We are interested in the asymptotics for the number of saddle connections when the bound L tends to infinity. In the model case of the torus it is sufficient to pass to the universal covering plane to see that the number of geodesic segments joining a generic pair of points grows quadratically: same as the volume of a ball, that is as πL^{2}.

It is proved in [EMa] that the growth rate of the number of saddle connections for a generic flat surface also has quadratic asymptotics c·(πL^{2}), and, moreover, almost all flat surfaces in some natural families of flat surfaces share the same constant c in the asymptotics. One of the principal goals of this paper is to compute these constants for all natural families (which are explicitly defined below). We also compute the constants in the quadratic asymptotics for the number of families of closed regular geodesics.

It is relatively easy to show that for a generic flat surface one can never find a pair of saddle connections having the same direction but different lengths. However, somewhat surprisingly, the configurations of pairs of parallel saddle connections of the same length and same direction can be found on almost every flat surface, and, what may seem even more surprising, their number also has quadratic asymptotics (with another constant c).

One can ask now the similar questions about the triples of saddle connections, or about some more specific configurations of saddle connections, or similar questions concerning closed geodesics. We give a complete description of those configurations which can be found on a generic surface; we show that each of them has quadratic asymptotics (this was essentially already proved in [EMa]), and in every case we compute the corresponding constant in the quadratic asymptotics.

Remark 1.2. A rational billiard table gives rise to such a flat surface. See for example [KaZe], [KMS], or [MT]. However, rational billiards are not generic in these natural families and so the constants described in this paper do not apply to the billiard problem, with an exception for some particular classes off billiards, see [EZo].

1.1. Flat Surfaces and Abelian Differentials. If we cut a flat surface S successively along an appropriate collection of saddle connections we can decompose it into polygons contained in ℝ^{2} = ℂ, see, for example, Figure 1. We may then view S as a union of polygons glued along parallel sides by means of parallel translations. Note that we have endowed every polygon with a complex coordinate. Since our gluing rules are just translations, the transition functions in these complex coordinates have the form

z → z + c.

Thus any flat surface with the conical singularities removed is endowed with a natural complex structure. Moreover, consider a holomorphic 1-form ω = dz on every polygon. Since dz = d(z + c) we obtain a globally defined holomorphic 1-form on the surface with removed singularities. It is an easy exercise to show that the complex structure and the holomorphic differential can be extended to the singularities; the holomorphic 1-form (or, what is the same, the Abelian differential) ω has a zero at every conical point.

Conversely, given a pair (M, ω) where M is a Riemann surface, ω is a holomorphic Abelian differential on M, and a point P ∈ M such that ω(P) ≠ 0, there exists a local coordinate z near P such that ω = dz. Such a local coordinate is unique up to change of coordinates z → z + c. Thus |dz|^{2} is a flat metric on M; this metric develops conical singularities at the zeroes of ω. At a zero of order k the total angle is 2π(k + 1). Furthermore, such a coordinate chart near every nonsingular point determines a structure S = (M, ω) of a “translation surface” (which we agree to call just a “flat surface”) on M, namely an atlas of coordinate charts which cover the surface away from the singularities, such that the transition functions are translations

z → z + c.

Specifying a direction means in this language that we distinguish between the flat surface defined by ω and e^{iθ}ω.

1.2. Holonomy and the sets V_{cyl}(S) and V_{sc}(S). Note that in terms of the local coordinates (x, y) defining the flat structure, ω = dz = dx + idy. This has an important consequence: For any oriented curve β on S, the holonomy of β (as defined above in terms of the developing map) coincides with the period ∫_{β} ω of ω over β, (and we have identified ℝ^{2} and ℂ). In particular, if β and γ are homologous, then hol(γ) = hol(β).

Let Cyl(S) denote the set of cylinders of regular closed geodesics on S. For a cylinder C ∈ Cyl(S), let hol(C) ∈ ℝ^{2} denote the holonomy of a regular closed geodesic γ in C. The vector hol(C) is independent of the choice of γ in C since all regular closed geodesics belonging to a given cylinder are homologous. We now define V_{cyl}(S) ⊂ ℝ^{2} as

V_{cyl}(S) = hol(Cyl(S)).

For example if S_{0} is the standard flat torus (i.e. the unit square with opposite sides identified), then V_{cyl}(S_{0}) = {(a, b) ∈ ℤ^{2} : gcd(a, b) = 1}.

In general V_{cyl}(S) is a discrete subset of ℝ^{2}. The number of cylinders on S of length at most L (counted without multiplicity) is exactly the cardinality of the intersection of V_{cyl}(S) with the ball of radius L centered at the origin.

We note that the map hol : Cyl(S) → V_{cyl}(S) need not be one to one (see Figure 2 for an example where there are two cylinders with the same holonomy). We define the multiplicity of a point v ∈ V_{cyl}(S) to be the number of distinct cylinders C such that hol(C) = v. Somewhat surprisingly, higher multiplicity is very common even for a generic surface. For example we will show that if the genus is at least 3, the number of multiplicity 2 vectors in V_{cyl}(S) of length at most L also grows quadratically in L.

We can repeat the same construction for the saddle connections. Let SC(S) denote the set of saddle connections on S, and let V_{sc}(S) = hol(SC(S)). Again, the map hol : SC(S) → V_{sc}(S) is far from one to one. For example, in Figure 2, there are four distinct saddle connections which map to the vector (1, 0). We will handle the multiplicites more carefully in §3.1.

2. Moduli Spaces of Abelian Differentials

2.1. Stratification. Let α be a partition of 2g − 2 (i.e. a representation of 2g − 2 as a sum of positive integers). Let H(α) denote the moduli space of pairs (M, ω) where M is a closed Riemann surface of genus g, and ω is a holomorphic Abelian differential on M such that the orders of its zeroes is given by α. The zeroes on each surface are assumed to be named. We say that two such structures are equivalent if there is an isomorphism from one to the other which takes zeroes to zeroes, preserving the naming. The set of equivalence classes is denoted H(α) and is called a stratum. This term is justified by the fact that the space of all Abelian differentials on Riemann surfaces of genus g is stratified by the spaces H(α), as α Figure 2. Consider four unit squares with sides glued as indicated. The result is a closed surface S of genus 2. The five points indicated by the filled circle symbol are identified, as are the five points indicated by the unfilled circle symbol. Hence S has two conical singularities, each of total angle 4π. Each of the shaded squares corresponds to a cylinder C_{i} of horizontal regular closed geodesics of length 1. Hence, hol(C_{1}) = hol(C_{2}) = (1, 0). Note, however, that the corresponding closed regular geodesics are not homologous. This means that the relation above would not survive under a generic deformation of this flat surface. Figure 9 gives an example when such relation is stable under any small deformation of the flat surface.

varies over the partitions of 2g − 2. The stratum corresponding to the partition (1, …, 1) is called the principal stratum: it corresponds to holomorphic differentials with simple zeroes. For example, the surface in Figure 2 belongs to H(1, 1). The stratum might be nonconnected, although the components have been completely classified (see [KonZo]). The classification involves the notion of parity of spin structure and issues of hyperellipticity. These components will be described later in the introduction.

2.2. Local Coordinates and Volume Element. The stratum H(α) can be topologized and given a natural “Lebesgue” measure as follows. For a flat surface S_{0} ∈ H(α) with conical points P_{1}, …, P_{n}, choose a basis of cycles for the relative homology H_{1}(S_{0}, {P_{1}, …, P_{n}}; ℤ). This basis may be chosen so that each element is represented by a saddle connection. Equivalently, the saddle connections cut S into a union of polygons. For any S near S_{0} the vectors associated to these saddle connections serve as local coordinates, see, for example, Figure 1. By construction these vectors are the relative periods of the Abelian differential ω (i.e. the integrals of ω along the paths joining points P_{i} and P_{j}, where i = j is also allowed). So, we, actually, use a domain in the space of relative cohomology H^{1}(S_{0}, {P_{1}, …, P_{n}}; ℂ) as a local coordinate chart. Ignoring the complex structure we get a domain in ℝ^{n} = H^{1}(S_{0}, {P_{1}, …, P_{n}}; ℂ). Note that we have a natural cubic lattice in our coordinates given by the “integer” cohomology

H^{1}(S_{0}, {P_{1}, …, P_{n}}; ℤ ⊕ iℤ) ⊂ H^{1}(S_{0}, {P_{1}, …, P_{n}}; ℂ) = ℝ^{n}

We define a measure dν(S) on H(α) as Lebesgue measure defined by these coordinates, normalized so that the volume of a unit cube in ℝ^{n} is 1. One easily checks that the measure is well-defined independent of choice of basis. Let H_{1}(α) ⊂ H(α) be the hypersurface in H(α) of unit area flat surfaces.

Choose a symplectic homology basis of closed curves A_{i}, B_{i}, i = 1, …, g on the surface S = (M, ω). The area of the flat surface S defined by the Abelian differential ω is given by the Riemann bilinear relation,

∫_{S} |ω|^{2}dxdy = \frac{i}{2} ∫_{S} ω ∧ ω̄ = \frac{i}{2} ∑_{i} (∫_{A_{i}} ω ∫_{B_{i}} ω̄ − ∫_{A_{i}} ω̄ ∫_{B_{i}} ω)

Thus the unit area surfaces represent a “hyperboloid” in terms of the coordinates chosen above.

Throughout this paper we shall mostly work with the “hyperboloid” H_{1}(α) ⊂ H(α). The volume element (and the measure) in the embodying space H(α) induce naturally the volume element (and the measure) on the hypersurface H_{1}(α) in the following way. There is a natural ℂ^{∗}-action on H(α): having λ ∈ ℂ − {0} we associate to the flat surface S = (M, ω) the flat surface λ · S = (M, λ · ω). In particular, we can represent any S ∈ H(α) as S = rS′, where r ∈ ℝ − {0}, and where S′ belongs already to the “hyperboloid”: S′ ∈ H_{1}(α). Geometrically this means that vectors associated to saddle connections on S′ are multiplied by r to give vectors on S. It means also that area(S) = r^{2}·area(S′) = r^{2}, since area(S′) = 1. We define the volume element d vol(S′) on the “hyperboloid” H_{1}(α) by disintegration of the volume element dν(S) on H(α):

dν(S) = r^{n−1} drd vol(S′) ,

where n = dim_{ℝ} H(α). Using this volume element we define the total volume of the stratum H_{1}(α):

(1)  Vol(H_{1}(α)) := ∫_{H_{1}(α)} d vol(S′) .

Since the measure on H_{1}(α) is induced by the volume element, the total volume of the stratum coincides with the total measure.

For a subset E ⊂ H_{1}(α) we let C(E) ⊂ H(α) denote the “cone” based on E:

(2)  C(E) := {S = rS′ | S′ ∈ E, 0 < r ≤ 1} .

Our definition of the volume element on H_{1}(α) is consistent with the following normalization:

(3)  Vol(H_{1}(α)) = n · ν(C(H_{1}(α)) ,

where n = dim_{ℝ} H(α), and ν(C(H_{1}(α)) is the total volume (total measure) of the “cone” C(H_{1}(α)) ⊂ H(α) measure by means of the volume element dν(S) on H(α) defined above.

It is a result of [M], [MS] and [Ve2] that these volumes are finite. The values have been calculated in [EOk], see also [Zo] for the values of the volumes of some low-dimensional strata. These will be needed for the calculations of the asymptotic constants. Tables of their values for various low genus examples are presented in section 6.4.

3. Evaluation of the Constants

3.1. Configurations of Saddle Connections. It is convenient to formalize the problem of counting saddle connections and the problem of counting closed geodesics in the following way.

For the first problem consider the set of vectors in ℝ^{2} associated to a saddle connection that starts at a zero of order m_{1} and ends at a distinct zero of order m_{2}. For some vectors there is a single saddle connection whose holonomy is that vector, and we say the vector has multiplicity one. There, however, may be a set of saddle connections connecting the two zeroes that map to the same vector under the holonomy map. In that case we say that the vector has higher multiplicity. It follows from the definition of the volume element that if a generic surface has saddle connections with higher multiplicity, then those saddle connections are homologous. Thus from now on we can assume the saddle connections are homologous in the case of higher multiplicity.

Now suppose that we have precisely p homologous saddle connections γ_{1}, …, γ_{p} joining z_{1} of order m_{1} to z_{2} of order m_{2}. All the γ_{i}, 1 ≤ i ≤ p have the same holonomy (where the orientation of each γ_{i} is from z_{1} to z_{2}). By convention the cyclic order of the γ_{i} at z_{1} is clockwise in the orientation defined by the flat structure. Let the angle between γ_{i} and γ_{i+1} at z_{1} be 2π(a′_{i} + 1); let the angle between γ_{i} and γ_{i+1} at z_{2} be 2π(a′′_{i} + 1). We call this the configuration

C = (m_{1}, m_{2}, a′_{i}, a′′_{i})

of the p homologous saddle connections. If there is just a single saddle connection (p = 1) then C = (m_{1}, m_{2}).

Remark 3.1. Since the flat structure is defined by a 1 form ω, and the holonomy of a curve β is given by ∫_{β} ω, two homologous curves always have the same holonomy. However as mentioned previously the geodesic representative of a homotopy class may consist of a broken line of saddle connections. Thus the higher multiplicity phenomenon occurs when homologous, nonhomotopic geodesics are each represented by a single saddle connection.

Now given a surface S and a configuration C, let V_{C}(S) denote the vectors v in ℝ^{2} such that precisely the p saddle connections in the configuration map to v under the holonomy map. We want to compute the asymptotics as T → ∞ of

|V_{C}(S) ∩ B(T)|,

where B(T) is the ball of radius T.

Now consider the second problem of this paper. We have a surface S and a saddle connection γ_{1} joining a zero to itself. For this problem we shall count the primitive elements. There may be other saddle connections γ_{2}, …, γ_{m} which are homologous to γ_{1}. Some of them may start and end at the same zero, the others may start and end at the other zeroes. Each γ_{i} returns to its zero at some angle θ_{i}, which must be an odd multiple of π. The return angles of π and −π are special. In these cases, the saddle connection bounds a cylinder. The other saddle connection bounding the cylinder may or may not be based at the same zero. For some vectors there is a single saddle connection whose holonomy is that vector, or pair if they bound a cylinder, and in either case, we say the vector has multiplicity one.

Suppose the curves γ_{i} bound q cylinders. The complement of the curves and cylinders are p = m − q surfaces. We say the multiplicity is p. The complement of the curves and cylinders are p = m − q surfaces S_{1}, …, S_{p}. We can associate a set J = {i_{1}, …, i_{q}} ⊂ {1, …, p}, where 0 ≤ q ≤ p with the property that if i_{j} ∈ J then there is a cylinder between γ_{i_{j}} and γ_{i_{j}+1}. By convention, when J = ∅ we let q = 0, that is we have no cylinders at all.

Now suppose a surface S_{i} is bounded by two homologous curves that leave and return to the same zero. The two curves bound angles 2(a′_{i} + 1)π and 2(a′′_{i} + 1)π at the zero. The two boundary circles are joined together at the zero to form a “figure eight”. If the two homologous curves that bound S_{i} return to different zeroes, then they bound angles b′_{i} and b′′_{i} at those zeroes. The boundary circles are disjoint in that case.

Altogether this gives a configuration C

C = (J, a′_{i}, a′′_{i}, b′_{k}, b′′_{k})

that are associated to this collection of saddle connections.

As in the first problem for a surface S and configuration C let V_{C} denote the vectors in ℝ^{2} such that the saddle connections in C have holonomy equal to that vector. Again we shall compute the asymptotics as T → ∞ of

|V_{C}(S) ∩ B(T)|,

where B(T) is the ball of radius T.

For either of the problems, given the configuration C, for each component of each H_{1}(α), there is a constant c = c(α, C) (depending on the connected component of H(α)) such that by [EMa] for almost all S in that component of H_{1}(α),

|V_{C}(S) ∩ B(T)| ∼ πcT^{2}.

Thus the growth rate is quadratic for a generic surface. What is perhaps surprising is that this formula says that the growth rate is quadratic for a generic surface even for multiple homologous saddle connections. By contrast, the definition of the measure ν implies that a generic surface does not have any pairs of saddle connections that determine vectors in the same direction but with different lengths.

Remark 3.2. The issue of higher multiplicity and specifying the angles is not explicitly addressed in [EMa]. However, the set V_{C}(S) defined above, satisfies all of the axioms of that paper, so the asymptotic formula from [EMa] is applicable to the problems formulated above.

One of the main goals of this paper is to evaluate the constant c. We shall have numerous formulae throughout the paper which will refer to the quadratic asymptotic constant computed above. In all cases they will hold for almost every surface in the component.

We also note that these constants have been verified numerically. A formula based on the formula of Kontsevich (see [Ko]) expresses the sum of the Lyapunov exponents of the Teichmüller geodesic flow on each connected component of every stratum H(α) in terms of some specific linear combination of the constants c(α, C) for the corresponding component of H(α). These Lyapunov exponents have been computed numerically and the constants achieved in this paper are consistent with these exponents and with Kontsevich’s formula.

3.2. Siegel–Veech Formula and Computation of the Constant. In [EMa] it was proved that for each configuration C, the constant on each component of H_{1}(α) satisfies the following Siegel–Veech formula:

(4)  \frac{1}{Vol(H_{1}(α))} ∫_{H_{1}(α)} f̂(S) d vol(S) = c ∫_{ℝ^{2}} f

where f is a continuous function of compact support in ℝ^{2} and

f̂(S) = ∑_{v⃗∈V_{C}(S)} f(v⃗).

The terms in the sum are counted with multiplicity; that is, we also count vectors associated to saddle connections that are not in the configuration if they are in the support of f. Thus a major part of this paper can be thought of as finding the constant in the Siegel–Veech formula. The strategy for evaluating c consists of letting f be the characteristic function of B(ϵ) and computing the leading term of order ϵ^{2} in the asymptotics of the left-hand-side of (4) as ϵ → 0. The support of f̂ is the space, denoted H_{1}^{ϵ}(α, C). This is the set of flat surfaces such that the saddle connections with the given configuration have length at most ϵ. This space is not compact, since as we let ϵ → 0, one leaves compact sets. The space H_{1}^{ϵ}(α, C) is also difficult to analyze since there may be saddle connections shorter than ϵ other than those in the configuration C.

Accordingly, let H_{1}^{ϵ,th}(α, C) denote the set of flat surfaces in H_{1}^{ϵ}(α) such that all saddle connections or closed geodesics that are not homologous to the ones in the configuration C have length greater than 2ϵ. (The explanation for the superscript “th” is that we think of these surfaces as lying in the part of the stratum which except for the curves in the configuration is “thick”.) The point of this definition is that f̂ is identically 1 on this space. The space will also prove to be easier to analyze since we shall be able to associate a boundary to it.

We shall find the asymptotics of ∫_{H_{1}^{ϵ,C}(α)} f̂(S) d vol(S) by breaking the domain

of integration H_{1}^{ϵ}(α, C) into H_{1}^{ϵ,th}(α) and its complement, the thin part H_{1}^{ϵ}(α, C) − H_{1}^{ϵ,th}(α, C). We now claim that

(5)  ∫_{H_{1}^{ϵ}(α,C)−H_{1}^{ϵ,th}(α,C)} f̂(S) d vol = o(ϵ^{2}).

To prove (5), we decompose H_{1}^{ϵ}(α, C) − H_{1}^{ϵ,th}(α, C) into a union of sets where a saddle connection or closed geodesic has length between (2ϵ)^{n+1} and (2ϵ)^{n}, where n is a positive integer. Since by definition, on each surface in this set there is a saddle with length between (2ϵ)^{n+1} and (2ϵ)^{n} and a nonhomologous saddle with length at most ϵ, the volume of that set is O(2^{n}ϵ^{2n+2}). We have the following estimate proved in [EMa] which is needed to bound the integrand. Let l(S) denote the length of the shortest saddle connection or closed geodesic on S. Then there are constants c′ and 0 < δ < 1 so that for S ∈ H_{1}^{ϵ}(α, C)

f̂(S) ≤ c′(l(S)^{−1−δ}),

These two estimates imply that the integral of f̂ on the set where there is a saddle of length between (2ϵ)^{n+1} and (2ϵ)^{n} is

O(ϵ^{2}2^{n}ϵ^{n(1−δ)})

Since δ < 1, summing over n, we find that (5) holds.

Then, since f̂ is zero outside H_{1}^{ϵ}(α, C) and f̂ = 1 on H_{1}^{ϵ,th}(α, C), we have

(6)  ∫_{H_{1}(α)} f̂(S) d vol = ∫_{H_{1}^{ϵ}(α,C)} f̂(S) d vol = Vol(H_{1}^{ϵ,th}(α, C)) + o(ϵ^{2}).

As we shall see, Vol(H_{1}^{ϵ,th}(α, C)) is always of the order ϵ^{2}, which is the order of the leading term that we are trying to find. Thus in all cases in this paper, after substituting (6) into (4) we see that

πc = lim_{ϵ→0} \frac{1}{ϵ^{2}} \frac{Vol(H_{1}^{ϵ,th}(α, C))}{Vol(H_{1}(α))}.

Thus the goal of this paper will be to calculate the asymptotics of the numerator. Now as one lets ϵ → 0, the flat surfaces in H_{1}^{ϵ,th}(α, C) degenerate to simpler surfaces. In the case of a single saddle connection joining a pair of distinct zeroes the zeroes coalesce to a higher order zero giving a surface in a stratum of higher codimension. If there are multiple saddle connections joining distinct zeroes, or as in the second problem there are closed geodesics, the result are flat surfaces of lower genus which may be highly degenerate. In any of the cases we shall say that the resulting surfaces belong to the principal boundary of the original stratum. Each configuration will determine a component of this principal boundary. A major part of this paper then will be to describe th is principal boundary, an object of independent interest.

Conversely, given surfaces in the principal part of the boundary, we will describe a set of constructions that allow us to recover surfaces in H_{1}^{ϵ,th}(α, C). The first is a splitting apart of a higher order zero into simpler zeroes that is the opposite of coalescing. The second is a slit and gluing construction which is the opposite of the degeneration of multiple saddle connections. To construct surfaces with homologous closed geodesics, we shall also need a pair of constructions, called the parallelogram and figure eight constructions.

In order to completely describe the way the configurations of saddles can arise from the principal boundary, we shall need to describe in combinatorial terms the number of ways these constructions can be performed. Because we are interested in components of strata we shall also need in certain situations to pay attention to parity of spin structures and hyperellipticity. Once this is done, the correspondence between configurations and boundary surfaces will allow us to compute Vol(H_{1}^{ϵ,th}(α, C)) in terms of the volumes of components of strata of these simpler surfaces and the combinatorial information about how many ways there are to build surfaces form the simpler surfaces. There will also be a normalizing factor which is the volume of the component of the stratum of the given H_{1}(α).

One can think of every stratum as an analogue of a complex polyhedron; the boundary of the polyhedron has “faces” of complex codimension 1, “edges” of complex codimension 2, etc. We may think of our calculations as follows. We could define a tubular ϵ-neighborhood of the boundary as the subset of those flat surfaces which contain a geodesic element of length at most ϵ. We could similarly define a tubular ϵ-neighborhood of the “edges” of the boundary as the subset of those flat surfaces which contain at least two nonhomologous geodesic elements of length at most ϵ. Thus our calculation shows that for a moderately growing function near the boundary, we can ignore the contribution in a tubular neighborhood of the edges.

Remark 3.3. One can consider these same problems in the context of quadratic differentials. To avoid overloading the paper we prefer to treat them in a subsequent one.

4. Readers Guide

In the first half of the paper we solve these problems for saddle connections joining distinct zeroes. We describe the multiplicity one case first, because it is the easiest and illustrates the computations. This is done in Sections 6.1–6.3. We then describe the principal boundary in the higher multiplicity case, describing the slit and gluing constructions and again give the constants. This is done in Sections 7.1–7.2 and 7.5. Sections 8.1–8.3 as well as the corresponding preliminaries from Section 5.1 are devoted to describing the constants in the cases of strata with several components and could be skipped by the reader interested only in the general thrust of the paper.

In the second half of the paper we consider the case of closed geodesics. We describe the principal boundary using the parallelogram and figure eight constructions. The reader interested only in the structure of the principal boundary may consult Sections 9–10. The computation of the combinatorial factor is performed in Section 10.5 and the computation of the constant c is given in Section 11.1. The rest of Section 11 is devoted to numerous examples and Section 12 is devoted to the special case of strata with several connected components.

We are illustrating all constructions by numerous examples. All possible strata and all possible configurations in genera g = 2 and g = 3 and all related computations are presented as the examples in the main body of the text. The complete list of admissible configurations in genus g = 4, and the values of the corresponding constants are presented in the Appendix. We also present in the Appendix the values of the constants for the two distinguished strata up to genus g = 30.

5. Connected Components of the Strata

5.1. Parity of a Spin Structure. As noted earlier, the strata H_{1}(α) are not necessarily connected. To classify the components one needs the notion of parity of spin structure. Consider a smooth simple curve β on the flat surface S which does not contain any zeroes. We define the index ind(β) ∈ ℕ of the vector field tangent to β to be the degree of the corresponding Gauss map: the total change of the angle between the vector tangent to the curve and the vector tangent to the horizontal foliation is 2π · ind(β).

Now take a symplectic homology basis {A_{i}, B_{i}}, i = 1, …, g, such that the intersection matrix has the canonical form: A_{i} ∘ A_{j} = B_{i} ∘ B_{j} = 0, A_{i} ∘ B_{j} = δ_{ij}, 1 ≤ i, j ≤ g. Though such basis is not unique, traditionally it is called a canonical basis. Consider a collection of smooth closed curves representing the chosen basis. Denote them by the same symbols A_{i}, B_{i}.

The parity of the spin structure ϕ(S) of the flat structure is defined to be

ϕ(S) := ∑_{i=1}^{g}(ind(A_{i}) + 1)(ind(B_{i}) + 1) (mod 2)

Now the index of a small loop around a single zero of order k is k + 1 so the parity is only defined if the zeroes have even order, which we now assume. It follows from the results of D.Johnson [J] that the parity of the spin structure ϕ(S) does not depend neither on the choice of representatives, nor on the choice of the canonical homology bases.

Moreover, it follows from the results of M.Atiyah [At] that ϕ(S) is invariant under continuous deformations of the flat structure of the translation surface, (see [KonZo] for details).

Since we do not need the notion of a spin structure itself, we do not discuss it in this paper; the reader may consider the notion “parity of the spin structure” as a single term. Note that it is applicable only to those flat structures ω which have set of zeros α containing even numbers only.

5.2. Connected Components of the Strata. The components are classified as follows (see [KonZo]).

Consider the general case, when g > 3, and α is different from two exceptional cases: α ≠ (2g − 2), α ≠ (g − 1, g − 1). Under these assumptions the stratum H_{1}(α) is connected whenever collection α contains an odd number. When g > 3, α ≠ (2g −2), α ≠ (g −1, g −1), and α has only even elements, there are exactly two connected components corresponding to even and odd spin structures. We shall denote them by H_{1}^{even}(α) and H_{1}^{odd}(α) correspondingly.

The strata H_{1}(2g − 2) and H_{1}(g − 1, g − 1) have in addition a special component — the hyperelliptic one. It consists of flat structures on hyperelliptic surfaces. In both cases the hyperelliptic involution τ sends the flat structure ω to −ω; for H_{1}(g − 1, g − 1) we also require that τ interchanges two zeroes of ω.

For g > 3 the stratum H_{1}(2g − 2) has three connected components: the hyperelliptic one — H_{1}^{hyp}(2g − 2), and two nonhyperelliptic components — H_{1}^{even}(2g − 2) and H_{1}^{odd}(2g − 2) corresponding to even and odd spin structures. For even g > 3 the stratum H_{1}(g − 1, g − 1) has two connected components: the hyperelliptic one, and the nonhyperelliptic one H_{1}^{nonhyp}(g − 1, g − 1). For odd g > 3 the stratum H_{1}(g − 1, g − 1) has three connected components: the hyperelliptic one, denoted by H_{1}^{hyp}(g − 1, g − 1), and two nonhyperelliptic ones — H_{1}^{even}(g − 1, g − 1) and H_{1}^{odd}(g − 1, g − 1) corresponding to even and odd spin structures.

Genera 2 and 3 are special. In genus 2, all flat structures are hyperelliptic, so each of the H_{1}(2) and H_{1}(1, 1) have one component which is hyperelliptic. In genus 3, only hyperelliptic flat structures may have even parity of the spin structure. Thus H_{1}(4) and H_{1}(2, 2) each have two connected components, the hyperelliptic one and the one with odd spin structure. The other strata in genus 3 are connected.

The flat surfaces from hyperelliptic strata have the following parity of the spinstructure (see [KonZo]). The parity of the spin structure determined by a flat structure from the hyperelliptic component S ∈ H_{1}^{hyp}(2g − 2) equals

(7)  ϕ(S) = [\frac{g + 1}{2}] (mod 2)

where the square brackets denote the integer part of a number. The parity of the spin structure determined by a flat structure ω from the hyperelliptic component ω ∈ H_{1}^{hyp}(g − 1, g − 1), for odd genera g equals

(8)  ϕ(S) = (\frac{g + 1}{2}) (mod 2) for odd g

5.3. Strata of Surfaces with Marked Points. In this paper we shall often consider the strata H_{1}(α) of surfaces S = (M, ω) where we not only fix the zeroes of the Abelian differential ω, but we also mark one or more regular points on the surface. Say, H_{1}(3, 1, 0) will denote the surfaces of genus g = 3 endowed with an Abelian differential with zeroes of orders 3 and 1 having one additional marked point (“zero of order 0 of the Abelian differential”).

Throughout this paper we assume that all zeroes and all marked points are always numbered. Thus a stratum with one or more marked points has the natural structure of a fiber bundle over the corresponding stratum without marked points with a direct product (minus diagonals) of several copies of the translation surface as a fiber, where the number of copies equals the number of marked points. For example, the universal curve H_{1}(3, 1, 0) fibers over H_{1}(3, 1) with a fiber S.

In particular, there is the following formula for the dimension of the strata with marked points. Let g be the genus and let |α| denote the number of entries in α. Suppose that 0 ∉ α. Then

(9) dim_{ℂ} H(α, \underbrace{0, …, 0}_{n}) = \begin{cases}dim_{ℂ} H(α) + n = 2g − 1 + |α| + n & when α ≠ ∅ \\ dim_{ℂ} H(∅) + n − 1 = n + 1 & when α = ∅ \end{cases}

(The case α = ∅ is exceptional, since by the homogeneity of the torus we can translate one of the marked points to any prescribed point.)

The natural measure on the stratum with marked points disintegrates in the product measure, where the measure along the fiber is just the Lebesgue measure on S (correspondingly product of several copies of S) induced by the flat metric on S, and the measure on the base is the natural measure on the corresponding stratum taken already without marked points.

Considering the volumes of the strata we confine ourselves to the subspaces H_{1}(α) of the strata for which the area of every surface S (measured in the flat metric) is equal to one. The observation above implies that volumes of the strata with marked points coincide with the volumes of the corresponding strata without marked points:

(10)  Vol(H_{1}(α)) = Vol(H_{1}(α, 0)) = Vol(H_{1}(α, 0, 0)) = …

Part 1. Saddle Connections Joining Distinct Zeroes

In this part we describe the possible configurations of saddle connections joining distinct zeroes and compute the constants in the corresponding asymptotics.

To separate the basic construction and numerous details we start with the easiest case. In the first section of this part we assume that the saddle connection joining a pair of zeroes P_{i}, P_{j} has multiplicity one, i.e., there are no other saddle connections in the same direction joining the same pair of zeroes. In this section we also assume that the translation surface under consideration belongs to a connected stratum.

In the second section of this part we consider the problem in full generality for the translation surfaces from connected strata. In particular we give the explicit general formula for the surfaces from the principal stratum.

In the last section of this part we treat the surfaces from the strata which are not connected.

6. Saddle Connections of Multiplicity One. Connected Strata

Throughout this section we assume that the flat surface S under consideration belongs to a connected stratum H_{1}(α).

6.1. Collapsing a Pair of Zeroes, Principal Boundary. Suppose we have a surface S ∈ H_{1}(α), a saddle connection of length at most ϵ joining distinct zeroes w_{1}, w_{2} of orders m_{1} and m_{2} and no saddle connection homologous to it. Let z_{1}, …, z_{l} be the other zeroes. Choose a basis of cycles in the relative homology group H_{1}(S, {w_{1}, w_{2}, z_{1}, …, z_{l}}; ℂ). This basis can be represented by a collection of curves β_{i} on the surface S. One of these curves, β_{0} is the saddle joining w_{1} and w_{2}. We suppose that all other curves except (at most one) miss w_{1}, w_{2}; the remaining curve (if it exists), say, β_{1} is not closed; it has w_{1} as an endpoint. (If w_{1}, w_{2} are the only zeroes of S, then all curves other than the saddle are closed and this β_{1} does not exist. One can construct a flat surface S′ by collapsing the saddle connection of length ϵ to a zero w of order m = m_{1} + m_{2}. If β_{1} exists then we can replace β_{1} with a curve joined to w. One can do this deformation while keeping the holonomy fixed along the entire basis other than β_{0}. All named zeroes on S not affected by the collapse are given the same names on S′. We may think of this as a Whitehead move on the foliation in direction γ. The resulting surface S′ belongs to a stratum H_{1}(α′). By construction the partition α′ is obtained from the partition α by replacing the pair of entries m_{1}, m_{2} by the entry m. The stratum H_{1}(α′) is in the closure of H_{1}(α) inside the moduli space of all flat structures on the surfaces of genus g. We say H_{1}(α′) is the principal boundary of H_{1}(α) corresponding to this configuration.

6.2. Breaking up a Zero. Conversely, suppose we are given a flat surface S′ defined by ω′, a zero w of order m ≥ 2, a pair of positive integers m_{1}, m_{2}, such that m = m_{1} + m_{2}, and a vector γ ∈ ℝ^{2} of length ϵ. Further suppose that S′ does not have any saddle connection or closed geodesic of length smaller than 2ϵ. Let w, z_{1}, …, z_{l} be the set of all zeroes of the flat structure on S′. Choose a basis of cycles in the relative homology group H_{1}(S′, {w, z_{1}, …, z_{l}}; ℂ). Let this basis be represented by a collection of curves β_{i} on the surface S′. We suppose that all curves except (at most one) miss the point w; the remaining curve (if it exists), say, β_{1} is not closed; it has w as an endpoint. (If w is the only zero on S′, then all the basic curves are closed, and β_{1} does not exist. )

We take a disc of radius ϵ about w that misses all other zeroes. We may break up the zero w of order m on S′ into two zeroes z′, z′′ of orders m_{1} and m_{2} correspondingly with a vector γ joining them constructing a flat surface S. Again we can describe this breakup as a Whitehead move on the foliation in direction γ.

If m_{1} ≠ m_{2}, the number of ways of effecting the breakup is 2m + 2. However we have \overrightarrow{z′z′′} = γ for only half of the resulting surfaces, i.e. for m + 1 ones; for another m + 1 surfaces we have \overrightarrow{z′′z′} = γ. If m_{1} = m_{2} = m/2 the number of the resulting surfaces is again m+1. for every such surface there are two ways to assign “names”

MODULI OF ABELIAN DIFFERENTIALS: THE PRINCIPAL BOUNDARY 17

\begin{matrix}ε & & ε & & 6π \\ 2δ & ε+δ & ε−δ & ε−δ & ε+δ & 2δ & ε−δ & ε−δ & ε+δ & ε+δ & ε+δ & ε+δ \end{matrix}

Figure 3. Breaking up a zero of degree 2 into two simple zeroes. Note that the surgery is local: we do not change the flat metric outside of the neighborhood of the zero.

z′, z′′ to the newborn zeroes of order m/2. This doubles the number of resulting surfaces with “named” zeroes. However we again have only m + 1 surfaces with \overrightarrow{z′z′′} = γ; for another m + 1 ones we have \overrightarrow{z′′z′} = γ. We again choose only those m + 1 surfaces for which we have \overrightarrow{z′z′′} = γ.

Again we let the curve which had the endpoint at w (if it was present) keep the corresponding endpoint at z′ during the deformation. We may do this in a manner that preserves the holonomy along the homology basis of S′.

We denote the assignment by

(S′, γ, m) → (S, m_{1}, m_{2})

We denote by H_{1}(α) the stratum which contains the resulting flat surface S. By construction the partition α is obtained from the partition α′ by replacing the entry m by two entries m_{1} and m_{2}.

Recall that all the zeroes on the surface S′ are “named”. For all the zeroes on the new surface S different from the zero which was just broken we keep the same “names” on S as their initial “names” on S′.

Now choose a simply connected subset of H_{1}(α′) of full volume and remove the set of flat surfaces with a saddle connection or closed geodesic of length at most 2ϵ.

Call the resulting set F′ ⊂ H_{1}(α′). We have

Vol(H_{1}(α′) − F′) = O(ϵ^{2})

Choose a homology basis that is valid for all S′ ∈ F′. Thus distinct S′_{1}, S′_{2} have different holonomy on some basis element. As in Section 3.1 let H_{1}^{ϵ,th}(α) ⊂ H_{1}(α) be the set of flat surfaces with a saddle connection of length at most ϵ joining the named zeroes and no other saddle connections of length smaller than 2ϵ. We have

Lemma 6.1. For γ a vector in ℝ^{2} of length at most ϵ, except for a set of S′ ∈ F′ of volume 0, there are precisely m + 1 surfaces in H_{1}^{ϵ,th}(α). that are the result of the assignment (S′, γ, m) → (S, m_{1}, m_{2}). Moreover every surface in H_{1}^{ϵ,th}(α) is the result of such an assignment.

Proof. We have addressed every issue except the statement that there are m + 1 surfaces that are the result of the assignment. First notice that if S′_{1} ≠ S′_{2}, the holonomy of the two resulting S_{1}, S_{2} built out of S′_{1}, S′_{2} differ on the homology basis. Thus if they are isomorphic, the isomorphism does not respect the relative homology basis. This implies that S′_{1}, S′_{2} must have a saddle connection of length at most ϵ. However we have removed surfaces in H_{1}(α′) with saddle connections or closed curves of length at most 2ϵ. The remaining obstacle is that for fixed S′, two or more of the m + 1 surfaces S built from S′ are isomorphic by an isomorphism that fixes the short saddle connection. But this isomorphism is induced by an isomorphism of S′ that fixes the zero, and the set of flat surfaces that have automorphisms has measure 0. □

Remark 6.2. Note that the cohomology class [ω] ∈ H^{1}(S, {z′, z_{1}, …, z_{l}}; ℂ) together with \overrightarrow{z′z′′} = γ determines an element of the relative cohomology group [ω] ∈ H^{1}(S, {z′, z′′, z_{1}, …, z_{l}}; ℂ). The lemma above claims, actually, that the mapping H_{1}(α) → H^{1}(S, {z′, z′′, z_{1}, …, z_{l}}; ℂ), defined by S → [ω] is a ramified covering of order m + 1 with a ramification locus at H_{1}(α′).

6.3. Computing the Asymptotic Constants. In this section we derive formulae for the constant c in quadratic asymptotics of the number of saddle connections joining a zero of order m_{1} to a zero of order m_{2} in multiplicity one. Recall from the introduction that this means that we have to find the asymptotics of Vol(H_{1}^{ϵ,th}(α)). Recall also that we are assuming that the zeroes are named. Since we may have numerous zeroes of orders m_{1} and m_{2} we, actually, have two different counting problems:

Problem 1. Count the constant in the quadratic asymptotics for the number of saddle connections joining a fixed zero z_{1} of order m_{1} to a fixed zero z_{2} of order m_{2}. (When m_{1} = m_{2} we require that z_{1} ≠ z_{2}.)

Problem 2. Count the constant in the quadratic asymptotics for the number of saddle connections joining some zero of order m_{1} to some other zero of order m_{2}. (This is, in fact, equivalent to counting the saddle connections joining unnumbered zeroes.)

Let us start by assuming that H_{1}(α) is connected. The case of strata that are not connected will be postponed until the end of the section on higher multiplicity.

Again let n = dim_{ℝ} H(α), n′ = dim_{ℝ} H(α′). Then n′ = n − 2. Recall that dν′(S′) is the natural measure on H(α′). For S′ ∈ C(F′) we set S′ = rS′′ where

[figure: 6π]

Figure 4. The cone angle corresponding to a zero of order m = 2 is equal to (m + 1) · 2π = 6π. Thus we have (m + 1) = 3 different ways of breaking up a zero of order 2 in a direction γ⃗. In this way (generically) we get m = 3 different flat surfaces

area(S′′) = 1 so area(S′) = r^{2}. Then by Lemma 6.1,

dν(S) = (m_{1} + m_{2} + 1)dν′(S′) dγ = (m_{1} + m_{2} + 1)r^{n′−1} dr d vol′(S′′) dγ

Again by Lemma 6.1,

ν(C(H_{1}^{ϵ,th}(α, C))) = (m_{1} + m_{2} + 1) Vol(F′) ∫_{0}^{1} r^{n′−1} ∫_{B(ϵr)} dγ dr

= πϵ^{2}(m_{1} + m_{2} + 1) Vol(F′) ∫_{0}^{1} r^{n′+1} dr.

= \frac{πϵ^{2}}{n′ + 2}(m_{1} + m_{2} + 1) Vol(F′).

Since n = n′ + 2 we have

Vol(H_{1}^{ϵ,th}(α, C)) = nν(C(H_{1}^{ϵ,th}(α))) = (m_{1} + m_{2} + 1)πϵ^{2} Vol(F′)

Thus, the constant from the Problem 1 has the following form:

c = lim_{ϵ→0} \frac{Vol(H_{1}^{ϵ,th}(α, C))}{πϵ^{2} Vol(H_{1}(α))} = lim_{ϵ→0} \frac{(m_{1} + m_{2} + 1) Vol(F′))}{Vol(H_{1}(α))} =

= \frac{(m_{1} + m_{2} + 1) Vol(H_{1}(α′))}{Vol(H_{1}(α))}.

Note that having solved the Problem 1 we get a solution of Problem 2 by an elementary combinatorial calculation. Let o(m_{i}) be the number of zeroes of order m_{i} in the stratum H(α). If m_{1} ≠ m_{2}, then there are o(m_{1}) ways of choosing the zero of order m_{1} and o(m_{2}) ways of choosing the zero of order m_{2}. Thus we get an additional factor o(m_{1}) · o(m_{2}) in comparison with Problem 1.

Formula 6.1. The constant in any connected stratum H_{1}(α) for the number of saddle connections of multiplicity one joining two zeroes of orders m_{1} ≠ m_{2} is equal to

(11)  c = o(m_{1})o(m_{2}) · (m_{1} + m_{2} + 1) · \frac{Vol(H_{1}(α′))}{Vol(H_{1}(α))} when m_{1} ≠ m_{2}

If m_{1} = m_{2}, then there are o(m_{1})(o(m_{1}) − 1)/2 ways of choosing an unordered pair of zeroes of order m_{1}. Having chosen an unordered pair, we choose, which of two zeroes would called be z_{1}, and which one would be z_{2} in arbitrary way: the number of saddle connections is obviously symmetric with respect to interchange of the points z_{1}, z_{2}. Thus, when m_{1} = m_{2} the answer to Problem 2 is given as the answer for Problem 1 multiplied by an additional factor o(m_{1})(o(m_{1}) − 1)/2.

Formula 6.2. The constant for the number of saddle connections of multiplicity one joining two distinct zeroes of the same orders m_{1} is equal to

(12)  c = \frac{o(m_{1})(o(m_{1}) − 1)}{2} · (2m_{1} + 1) · \frac{Vol(H_{1}(α′))}{Vol(H_{1}(α))}  when m_{1} = m_{2}

Note that we may view the entire combinatorial calculation in terms of the naming of the zeroes. We can assume that on S′ we have broken up the zero of m = m_{1} + m_{2} at the first “named” zero of order m. The remaining zeroes are then given the same names on S,. Other zeroes of orders m_{1}, m_{2} already have names on S. There are o(m_{1})o(m_{2}) ways of adding additional names to zeroes of orders m_{1}, m_{2} if m_{1} ≠ m_{2} and o(m_{1})(o(m_{1}) − 1) if m_{1} = m_{2}.

Note that in the consideration above the stratum H_{1}(α′) might be nonconnected. In this case Vol(H_{1}(α′)) is the sum of the volumes of connected components. For example if we start with the stratum H_{1}(5, 3), then H_{1}(α′) has three connected components H_{1}^{hyp}(8), H_{1}^{even}(8) and H_{1}^{odd}(8), and the sum of the volumes of these must be in the numerator.

6.4. Examples: Constants for Connected Strata in Genus 3. As an illustration of the formulae above we now give the explicit values for these constants in a number of cases.

In Table 1 we present the normalized volumes \frac{1}{π^{2g}} Vol(H_{1}(α)) of the strata with numbered zeroes, see [EOk], see also some values in [Zo]. These values will be used for computations in all the examples.

Both strata in genus g = 2 are hyperelliptic; they will be treated later on in section 8. In genus g = 3 there are three connected strata: H_{1}(3, 1), H_{1}(2, 1, 1), and H_{1}(1, 1, 1, 1).

Example 6.3. Stratum H_{1}(3, 1). After collapsing zeroes we obtain a flat surface S′ ∈ H_{1}(4), where both components H_{1}^{hyp}(4) and H_{1}^{odd}(4) occur. We have m_{1} = Table 1. Normalized volumes \frac{1}{π^{2g}} Vol(H_{1}(α)) of the strata in small genera

Genera g = 1, 2
H_{1}(∅) H_{1}(2) H_{1}(1, 1)
\frac{1}{3}  \frac{1}{120}  \frac{1}{135}

Genus g = 3
H_{1}^{hyp}(4) H_{1}^{odd}(4) H_{1}(3, 1) H_{1}^{hyp}(2, 2) H_{1}^{odd}(2, 2) H_{1}(2, 1, 1) H_{1}(1, 1, 1, 1)
\frac{1}{6720}  \frac{1}{2430}  \frac{16}{42525}  \frac{1}{9450}  \frac{1}{4320}  \frac{1}{3780}  \frac{1}{4860}

Genus g = 4
H_{1}^{hyp}(6)  H_{1}^{odd}(6)  H_{1}^{even}(6)  H_{1}(5, 1)
\frac{1}{580608}  \frac{1}{37800}  \frac{32}{1913625}  \frac{1}{36750}

H_{1}^{odd}(4, 2)  H_{1}^{even}(4, 2)  H_{1}(4, 1, 1)  H_{1}^{nonhyp}(3, 3)
\frac{1}{79380}  \frac{1}{107520}  \frac{11}{653184}  \frac{1}{51030}

H_{1}^{hyp}(3, 3)  H_{1}(3, 2, 1)  H_{1}(3, 1, 1, 1)  H_{1}^{odd}(2, 2, 2)
\frac{1}{992250}  \frac{1}{70875}  \frac{62}{5740875}  \frac{31}{4354560}
H_{1}^{even}(2, 2, 2) H_{1}(2, 2, 1, 1) H_{1}(2, 1, 1, 1, 1) H_{1}(1, …, 1)
\frac{37}{6804000}  \frac{131}{13608000}  \frac{1}{136080}  \frac{377}{67359600}

3, m_{2} = 1, thus the combinatorial factor equals o(m_{1})o(m_{2})(m_{1} + m_{2} + 1) = 5. We get

c = 5 · \frac{Vol(H_{1}(4))}{Vol(H_{1}(3, 1))} = 5 · \frac{(Vol(H_{1}^{hyp}(4)) + Vol(H_{1}^{odd}(4))}{Vol(H_{1}(3, 1))} = \frac{7625}{1024} ≈ 7.45

Example 6.4. Stratum H_{1}(2, 1, 1). There are two cases here in multiplicity 1. The first case is a saddle connection joining the zero of degree 2 with any of the two zeroes of degree 1. Collapsing the zeroes we obtain a surface S′ ∈ H_{1}(3, 1). For m_{1} = 2, m_{2} = 1, o(m_{1}) = 1, o(m_{2}) = 2. Thus the combinatorial constant equals 8. We get

c = 8\frac{Vol(H_{1}(3, 1))}{Vol(H_{1}(2, 1, 1))} = \frac{512}{45} ≈ 11.4

The second case consists of collapsing the pair of simple zeroes to a zero of order 2. We obtain a surface S′ ∈ H_{1}(2, 2). Any surface in any component of H_{1}(2, 2) can be found by collapsing a pair of simple zeroes in this manner. Here m_{1} = m_{2} = 1 and o(m_{1}) = 2 so the combinatorial constant is 3. We get

c = 3 · \frac{Vol(H_{1}(2, 2))}{Vol(H_{1}(2, 1, 1))} = 3 · \frac{Vol(H_{1}^{hyp}(2, 2)) + Vol(H_{1}^{odd}(2, 2))}{Vol(H_{1}(2, 1, 1))} = \frac{153}{40} ≈ 3.83

6.5. Constants for the Principal Stratum H_{1}(1, …, 1). The surface of genus g has 2g − 2 simple zeroes; o(1) = 2g − 2. The corresponding surface S′ ∈ H_{1}(2, 1, …, 1) has 2g − 4 simple zeroes. The factor o(m_{1}) (o(m_{1}) − 1) /2 gives (g − 1)(2g − 3), and the factor (2m_{1} + 1) gives 3. Applying formula (12)) we get

Formula 6.3. The constant for the number of saddle connections of multiplicity one joining two distinct zeroes is equal to

c = 3(g − 1)(2g − 3) · \frac{Vol(H_{1}(2, \overbrace{1, …, 1}^{2g−4}))}{Vol(H_{1}(\underbrace{1, …, 1}_{2g−2}))}

In the table below we present some values of the constant in this case.

Table 2. Principal stratum H_{1}(1, …, 1); values of the constants for saddle connections of multiplicity one joining distinct zeroes

g = 2 g = 3 g = 4  g = 5  g = 6  g = 7  g = 8
c =  \frac{27}{8}  \frac{162}{7}  \frac{22275}{377} \frac{2594700}{23357} \frac{2954056635}{16493303} \frac{13229971542}{50280671} \frac{14740938123723}{40593663941}
c ≈ 3.375 23.14 59.08  111.1  179.1  263.1  363.1

7. Multiple Homologous Saddle Connections. Connected Strata

7.1. Principal Boundary. Consider a surface S ∈ H_{1}(α) with a fixed pair of zeroes z_{1} and z_{2} of orders m_{1} and m_{2} correspondingly. Suppose that we have a configuration C (see Section 3) of precisely p homologous saddle connections γ_{1}, …, γ_{p} joining z_{1} to z_{2}. A pair γ_{i} and γ_{i+1} bounds a surface S_{i}. The surfaces S_{i} and S_{i+1} share the saddle connection γ_{i+1}. By convention the cyclic order of γ_{i} at z_{1} is clockwise in the orientation defined by the flat structure. The angle between γ_{i} and γ_{i+1} at z_{1} be 2π(a′_{i} + 1); the angle between γ_{i} and γ_{i+1} at z_{2} be 2π(a′′_{i} + 1).

Cut the surface S along all γ_{i}, i = 1, …, p. Gluing together the two sides γ_{i} and γ_{i+1} of the boundary of S_{i} we get a flat surface with two distinguished zeroes z′_{i}, z′′_{i} of orders a′_{i} and a′′_{i} correspondingly joined by a saddle connection, \overrightarrow{z′_{i}z′′_{i}} = γ. When one of a′_{i}, a′′_{i} is equal to zero the corresponding point is just a marked point. If a′_{i} = a′′_{i} = 0, then both points are marked points.

By construction any resulting flat surface S_{i} has a saddle connection γ_{i} joining the pair of zeroes and does not have any other saddle connections homologous to it. This implies that contracting the saddle connection γ_{i} to a point we get a flat surface S′_{i} with a distinguished zero (marked point) w_{i} of order a_{i} = a′_{i} + a′′_{i}.

Let S′_{i} ∈ H(α′_{i}). We say that H(α′_{i}) is the principal boundary of this configuration.

7.2. Slit Construction. We would like to reverse the above degeneration and build surfaces with multiple homologous saddle connections out of surfaces in H(α′_{i}). We need the following construction. Let S′ be a surface with a zero of order a ≥ 0. Suppose there are no saddle connections of length shorter than 2ϵ. Let a′, a′′ ≥ 0 such that a′ + a′′ = a. Let γ be a vector of length ϵ. If a′, a′′ ≠ 0 we may break up the zero of order a into zeroes of orders a′, a′′ with a saddle connection determining γ joining them as before. If a′ = 0 take a point z′ of the form w − γ (this means along a geodesic from w in direction −γ and distance |γ|) and join it to z′′ = w on S′. If a′′ = 0 we take a segment from z′ = w to z′′ = w + γ on S′.

In either case we may slit the resulting surface along the saddle connection joining the two zeroes z′, z′′. In this way we build a surface with one boundary component consisting of two arcs, denoted γ′ and γ′′, joining the endpoints of the slit. The angles between γ′ and γ′′ at the points z′ and z′′ are 2π(a′ + 1) and 2π(a′′ + 1) correspondingly.

Note that the flat structure on S′ fixes the choice of the orientation. By convention we give the “names” γ′ and γ′′ to the arcs in such a way that turning around z′ in a clockwise direction from γ′′ to γ′ we do not leave the surface.

Conversely, consider a flat surface with a single boundary component consisting of two arcs joining a pair of vertices. The convention on the choice of the “names” γ′ and γ′′ means that as soon as we know which of two vertices is z′, and which is z′′ we can determine which of two arcs is γ′, and which is γ′′.

We describe now how to build surfaces with multiple homologous saddle connections using the slit construction.

7.3. Building Surfaces With Multiple Saddle Connections. Suppose S′ = S′_{1}, ⊔ …⊔S′_{p} is a disconnected flat surface and on each S′_{i} there is a zero or a regular point which we think of as a marked point w_{i} of order a_{i}. Given pairs a′_{i}, a′′_{i} with a_{i} = a′_{i} + a′′_{i}, and a vector γ, we perform the slit construction on each surface. We obtain surfaces with one boundary component each of which consists of two arcs γ′_{i} and γ′′_{i}. We glue γ′_{i} to γ′′_{i+1}, calling this curve γ_{i}, (and γ′_{p} to γ′′_{1}). This gives a closed surface S, a pair of zeroes z_{1} and z_{2} of orders m_{1} and m_{2}, where ∑ a′_{i} = m_{1} + 1 + p and ∑ a′′_{i} = m_{2} + 1 − p, and a set of homologous curves γ_{i}, i = 1, …, p joining z_{1} to z_{2}. The angle between γ_{i−1} and γ_{i} at z_{i} is 2π(a′_{i} + 1) and the angle between them at z_{2} is 2π(a′′_{i} + 1). We denote this assignment by

(S′, γ, a′_{i}, a′′_{i}) → (S, m_{1}, m_{2}, a′_{i}, a′′_{i})

Our convention on the choice of γ′_{i}, γ′′_{i} (see the section above) implies that the cyclic order ⋯ → γ_{i−1} → γ_{i} → γ_{i+1} → … is clockwise at z_{1} with respect to the orientation defined by the flat structure.

Fix a basis for the relative homology on each surface S′_{i}. A relative homology basis for S is given by:

• A relative homology basis of curves β_{i} for S′.

• A curve connecting the two zeroes. The integral of ω along this curve is γ. We introduce ᾱ′_{i} as follows:

α′_{i} = \begin{cases}ᾱ′_{i} ⊔ \{a_{i}\} & when a_{i} ≠ 0 \\ ᾱ′_{i} & when a_{i} = 0 \end{cases}

Having a collection of integers α we denote by |α| their sum, say, |α| = 2g − 2.

S_{3}  _{γ′_{3}} _{γ{′′ \atop 3}} _{γ′_{2}} _{γ{′′ \atop 2}}  S_{2}_{S_{1}_{γ{′′ \atop 1}} _{γ′_{1}}}  _{S_{3}  S_{2} z′′ z′}_{S_{1} S_{3}  S_{2} z_{2}_{γ_{1}}_{γ_{3}}}

Figure 5. Multiple homologous saddle connections.

The construction above implies the following conditions on partitions α′_{i} and the numbers a′_{i}, a′′_{i}.

Lemma 7.1. An assignment (S′, γ, a′_{i}, a′′_{i}) → (S, m_{1}, m_{2}, a′_{i}, a′′_{i}) satisfies the following necessary conditions on the collection (α′_{i}, a′_{i}, a′′_{i}):

a′_{1} + ⋯ + a′_{p} = m_{1} + 1 − p
a′′_{1} + ⋯ + a′′_{p} = m_{2} + 1 − p
ᾱ′_{1} ⊔ ⋯ ⊔ ᾱ′_{p} ⊔ {m_{1}} ⊔ {m_{2}} = α
|ᾱ′_{i}| = a′_{i} + a′′_{i} (mod 2) for i = 1, …, p

Moreover, when the stratum H_{1}(α) is connected, these conditions are sufficient: every surface S ∈ H_{1}(α) with a configuration of p homologous saddle connections joining the pair of zeroes can be obtained by an assignment (S′, γ, a′_{i}, a′′_{i}) → (S, m_{1}, m_{2}, a′_{i}, a′′_{i}) with appropriate (S′, γ, a′_{i}, a′′_{i}).

We shall introduce notation to describe the degeneration pattern for the surfaces S_{i}.

Notation. The integer represented as a sum of two integers corresponds to a_{i} represented as a′_{i} + a′′_{i}. Here the order of appearances of the summands a′_{i} + a′′_{i} is significant. The cyclic order of the surfaces is represented by S_{1} ≻ S_{2} ≻ S_{3} ≻ … ≻ S_{p} ≻ S_{1}.

Example 7.2. Let S ∈ H_{1}(4, 3, 2, 1); m_{1} = 3, m_{2} = 4; p = 3. There are 15 possible pictures for the three homologous saddle connections joining the zero of order 3 to the zero of order 4 depending on the returning angles (Table 3) Thus for example, the first line in table 3 refers to 3 surfaces in cyclic order. The first is a torus (no zeroes) where we have broken up a marked point into (0 + 0); the second surface

Degeneration pattern  α′_{1} α′_{2}  α′_{3}
(0 + 0) ≻ (1 + 1) ≻ (0 + 1, 2, 1) ≻ (∅) (2) (2, 1, 1)
(1 + 1) ≻ (0 + 0) ≻ (0 + 1, 2, 1) ≻ (2) (∅) (2, 1, 1)
(0 + 0) ≻ (0 + 0) ≻ (1 + 2, 2, 1) ≻ (∅) (∅) (3, 2, 1)
(0 + 2) ≻ (0 + 0) ≻ (1 + 0, 2, 1) ≻ (2) (∅) (2, 1, 1)
(0 + 0) ≻ (0 + 2) ≻ (1 + 0, 2, 1) ≻ (∅) (2) (2, 1, 1)
(1 + 1) ≻ (0 + 1, 1) ≻ (0 + 0, 2) ≻ (2) (1, 1) (2)
(0 + 0) ≻ (1 + 0, 1) ≻ (0 + 2, 2) ≻ (∅) (1, 1) (2, 2)
(0 + 2) ≻ (1 + 0, 1) ≻ (0 + 0, 2) ≻ (2) (1, 1) (2)
(0 + 0) ≻ (1 + 2, 1) ≻ (0 + 0, 2) ≻ (∅) (1, 3) (2)
(0 + 0) ≻ (0 + 1, 1) ≻ (1 + 1, 2) ≻ (∅) (1, 1) (2, 2)
(1 + 1) ≻ (0 + 0, 2) ≻ (0 + 1, 1) ≻ (2) (2) (1, 1)
(0 + 0) ≻ (1 + 1, 2) ≻ (0 + 1, 1) ≻ (∅) (2, 2) (1, 1)
(0 + 0) ≻ (0 + 0, 2) ≻ (1 + 2, 1) ≻ (∅) (2) (1, 3)
(0 + 2) ≻ (0 + 0, 2) ≻ (1 + 0, 1) ≻ (2) (2) (1, 1)
(0 + 0) ≻ (0 + 2, 2) ≻ (1 + 0, 1) ≻ (∅) (2, 2) (1, 1)

Table 3. Possible degenerations for S ∈ H_{1}(4, 3, 2, 1) with m_{1} = 3, m_{2} = 4; p = 3

has a single zero of order 2 which we have broken into two simple zeroes indicated by (1 + 1), and the third surface has zeroes of 2, 1, 1, where we have broken one of the simple zeroes into a simple zero and a zero of order 0.

7.4. Stratum Interchange and γ → −γ Symmetry. In this section we discuss the possible symmetries of the assignment

(S′, γ, a′_{i}, a′′_{i}) → (S, m_{1}, m_{2}, a′_{i}, a′′_{i})

Let us first specify our problem.

Problem 1. We assume that the zeroes z_{1}, …, z_{l} of the surface S are numbered. We fix the zeroes z_{1}, z_{2} ∈ S, z_{1} ≠ z_{2}, of orders m_{1} and m_{2} correspondingly.

We start with the case when we fix also the following data. We assume that the zeroes z_{3}, …, z_{l_{1}} belong to S_{1}, the zeroes z_{l_{1}+1}, …, z_{l_{1}+l_{2}} belong to S_{2}, ..., the zeroes z_{l_{1}+···+l_{p−1}+1}, …, z_{l_{1}+···+l_{p}} belong to S_{p}. Here l_{i} + 1 = card(α′_{i}), when a_{i} ≠ 0, and l_{i} = card(α′_{i}), when a_{i} = 0. We have l = l_{1} + ⋯ + l_{p} = card(α).

In this setting the only thing that is not determined is the case when S has exactly two zeroes: z_{1} and z_{2}. Then every S′_{i} either has a single zero of order a_{i} or it is a torus with a marked (regular) point. Note that we fix only the cyclic order of the collection (a′_{i}, a′′_{i}), but not the numbering of each of these pair of numbers. The natural action of the cyclic group of order p on the collection of ordered pairs (a′_{i}, a′′_{i}) organized in a cyclic order, may have nontrivial stabilizer, which we denote by Γ. We get symmetry of order |Γ|. This type of symmetry we call the stratum interchange.

Problem 2. Now consider the problem with fewer constrains. We count the number of occurrences of the following phenomenon: some zero of S of order m_{1} is joined to some zero of order m_{2} by precisely p homologous saddle connections γ_{1}, …, γ_{p} as above. We fix the type α′_{i} of the flat surface S′_{i} obtained as described above, but we make no assumptions on the distribution of the zeroes z_{1}, …, z_{l} by the surfaces S′_{i}. As before we fix only the cyclic order of appearances of the surfaces S_{i}.

Remark 7.3. Problem 2 may be considered as generalization of Problem 1 to the case when the zeroes of S and of S′_{i} are not numbered.

Remark 7.4. In both settings the H_{1}(α′_{i}) might be nonconnected. We could also specify the connected component of the H_{1}(α′_{i}) when this occurs. We prefer the setting where this data is not specified.

Consider the natural action of the cyclic group of order p on the collection

(α′_{1}, a′_{1}, a′′_{1}) ≻ ⋯ ≻ (α′_{p}, a′_{p}, a′′_{p}) ≻

organized in a cyclic order. If it has a nontrivial stabilizer we denote it by the same symbol Γ. We get a symmetry of order |Γ| which we also call the stratum interchange.

Example 7.5. Let α = (27, 15, 1, 1), m_{1} = 27, m_{2} = 15. Consider the degeneration of S into 6 surfaces of the following types α′_{1} ≻ ⋯ ≻ α′_{6} ≻

(2) ≻ (7, 1) ≻ (12) ≻ (2) ≻ (7, 1) ≻ (12) ≻

The possible symmetry in this case is a symmetry of order 2, where subgroup Γ is a shift by 3.

The degeneration

(0 + 2) ≻ (4 + 3, 1) ≻ (7 + 5) ≻ (0 + 2) ≻ (4 + 3, 1) ≻ (7 + 5) ≻

possesses this symmetry, while the degeneration

(0 + 2) ≻ (3 + 4, 1) ≻ (7 + 5) ≻ (2 + 0) ≻ (4 + 3, 1) ≻ (7 + 5) ≻

does not.

When m_{1} = m_{2} we may have an additional symmetry in Problem 2. Let P, Q be two zeroes of ω of order m_{1} = m_{2} joined by precisely p homologous saddle connections. Let the homology class of the saddle connection be represented by a vector γ. Assigning the “names” z_{1} := P and z_{2} := Q to P and Q we get a decomposition (S′_{i}, γ, α′_{i}, a′_{i}, a′′_{i}).

Consider now the very same surface S with the same configuration of p homologous saddle connections joining the same two zeroes P and Q. We may declare now that the correspondence of zeroes is inverse with respect to the previous assignment: z_{1} := Q and z_{2} := P. Since our saddle connections are oriented (from z_{1} to z_{2}) the homology class of the same saddle connection (from P to Q) is represented now by the vector −γ. Since the cyclic order of “pieces” is determined by the cyclic order at the point z_{1} the new identification reverses the cyclic order in the collection (α′_{i}), as well as the order in the pairs (a′_{i}, a′′_{i}).

We say, that we have a γ → −γ symmetry if and only if the assignment z_{1} := Q and z_{2} := P gives a decomposition with the same (up to a cyclic reenumeration) collection (α′_{i}, a′_{i}, a′′_{i}) as before.

It is easy to see the following combinatorial criterion of the γ → −γ symmetry. Interchange simultaneously all a′_{i} ↔ a′′_{i}, and change the cyclic order of the resulting collection (α′_{i}, a′′_{i}, a′_{i}) to the opposite one. We possess a γ → −γ symmetry if and only if the result of this operation gives us the initial collection (α′_{i}, a′_{i}, a′′_{i}) up to a cyclic reenumeration. We denote γ → −γ symmetry by Γ_{−}, where |Γ_{−}| = 2. We let |Γ_{−}| = 1 when there is no γ → −γ symmetry.

The condition m_{1} = m_{2} is an obvious necessary condition for γ → −γ symmetry. It will be always assumed in the discussions of γ → −γ symmetry.

Under this condition every degeneration of multiplicity 1 possesses a γ → −γ symmetry. There is a γ → −γ symmetry in multiplicity 2 if and only if at least one the following conditions is valid: either

— a′_{1} = a′′_{1} (and then a′_{2} = a′′_{2} automatically); or

— α′_{1} = α′_{2} and a′_{1} = a′′_{2} (and then a′′_{1} = a′_{2} automatically).

Example 7.6. Let α = (11, 11). Consider degeneration of S into 4 surfaces of the following types α′_{1} ≻ α′_{2} ≻ α′_{3} ≻ α′_{4}:

(2) ≻ (6) ≻ (2) ≻ (6) ≻

The degeneration

(0 + 2) ≻ (3 + 3) ≻ (2 + 0) ≻ (3 + 3) ≻

possesses the γ → −γ symmetry, but does not have the stratum interchange, so |Γ_{−}| = 2, |Γ| = 1.

The degeneration

(0 + 2) ≻ (4 + 2) ≻ (0 + 2) ≻ (4 + 2) ≻

does not have a γ → −γ symmetry, but allows the stratum interchange, so |Γ_{−}| = 1, |Γ| = 2.

The degeneration

(1 + 1) ≻ (3 + 3) ≻ (1 + 1) ≻ (3 + 3) ≻

has both symmetries: |Γ_{−}| = 2, |Γ| = 2.

Remark 7.7. Describing the assignments (S′, γ, a′_{i}, a′′_{i}) → (S, m_{1}, m_{2}, a′_{i}, a′′_{i}) one can consider all possible assignments, and then take into account corresponding symmetries, or one can deal with the classes, eliminating the symmetry whenever it is possible.

We have chosen the second way. For example the assignment

(2 + 0) ≻ (0 + 0) ≻ (0 + 1, 2, 1)

is not presented in the Table 3 since it is symmetric to the assignment

(0 + 0) ≻ (0 + 2) ≻ (1 + 0, 2, 1)

(which is in the list) by composition of γ → −γ symmetry with the stratum interchange.

As in the single saddle connection situation for small ϵ, we let F′ be a simply connected domain in H_{1}(α′) such that for every surface in F′, each saddle connection and closed geodesic has length at least 2ϵ. Again Vol(F′) differs from Vol(H_{1}(α′)) by O(ϵ^{2}). Let C(F′) the cone over F′. Let H_{1}^{ϵ,th}(α, C) be the subset of H_{1}(α) consisting of those (S, m_{1}, m_{2}, a′_{i}, a′′_{i}) with the prescribed set of homologous curves joining two zeroes of length at most ϵ and no other saddle connections or closed geodesic has length smaller than 2ϵ.

Lemma 7.8. For almost all S′ ∈ F′ and γ a vector in ℝ^{2} of length at most ϵ, there are exactly ∏_{i=1}^{p}(a_{i} + 1) flat surfaces S that are the result of the assignment (S′, γ, a′_{i}, a′′_{i}) → (S, m_{1}, m_{2}, a′_{i}, a′′_{i}), where a_{i} = a′_{i} + a′′_{i}. There are precisely 2|Γ| assignments that result in the same flat surface if there is a γ → −γ symmetry, |Γ| to 1 otherwise. The assignment is onto C(H_{1}^{ϵ,th}(α)).

Proof. We may choose a homology basis valid for all of F′. For distinct S′ and S′′ the holonomy must differ for some basis curve. The surfaces that correspond to S′ and S′′ by the assignment can be isomorphic only if the isomorphism does not preserve the homology basis. If the curves in the configuration are preserved, then the possibilities are that we have the γ → −γ symmetry, or we have the stratum symmetry. If the set of configurations curves are not preserved then S′ contains a curve of length at most ϵ in length. However the last possibility has been ruled out by the choice of F′. The only remaining possibility is that for fixed S′ two or more of the resulting S are isomorphic by an isomorphism that preserves the set of homologous curves. But such an isomorphism must come from an automorphism of S′ and the set of such S′ has measure 0. For each pair a′_{i}, a′′_{i} there are a_{i} + 1 ways to perform a slit construction at a zero of order a_{i} = a′_{i} + a′′_{i}, providing a_{i} + 1 surfaces. The Lemma now follows from Lemma 7.1.

□

7.5. The Constant in Higher Multiplicity, Connected Strata. We start with the study of the constant c in the setting where we fix the partition of numbered zeroes of S by components S′_{i} (see Problem 1 in section 7.4). We assume first that H_{1}(α) is connected.

By Lemma 7.8 the data

(S′, γ, a′_{i}, a′′_{i})

defines

M = \frac{1}{|Γ_{−}| · |Γ|} · ∏_{i=1}^{p}(a_{i} + 1)

surfaces S in the setting of Problem 1 of section 6.2.

We write S_{i} = r_{i}S′_{i}, where area S′_{i} = 1. Then area S_{i} = r_{i}^{2}.

For the computations in this section and the section on saddles joining a zero to itself, we shall use the following notation. By (9) we get

(13)  d_{i} = \begin{cases}n_{i} = dim_{ℝ} H_{1}(α′_{i}) & when a_{i} ≠ 0 or when α′_{i} = ∅ \\ dim_{ℝ} H_{1}(α′_{i}) + 2 & when α′_{i} ≠ ∅ and a_{i} = 0 \end{cases}

In case there are no marked points, we have

dν(S) = M ∏_{i=1}^{p} dν′_{i}(S_{i}) dγ = M ∏_{i=1}^{p} (r_{i}^{n_{i}−1} dr_{i}) ∏_{i=1}^{p} d vol′(S′_{i})

In case z_{i} is a marked point (i.e. a_{i} = 0), the formula for dν(S) is multiplied by dz_{i} since dz_{i} gives the area element of the varying point z_{i}. In that case the last form for the measure has an extra factor of r_{i}^{2}. Set

W = ∏_{i=1}^{p} Vol(H_{1}(α_{i})).

Then, in all cases up to an error term that is O(ϵ^{2}),

ν(C(H_{1}^{ϵ,th}(α, C)) = MW ∫_{D(1)} ∏_{j=1}^{p} r_{j}^{d_{j}−1} ∫_{B(ϵ\sqrt{∑ r_{j}^{2}})} dγ ∏_{j=1}^{p} dr_{j}

= MW ∫_{D(1)} ∏_{j=1}^{p} r_{j}^{d_{j}−1}πϵ^{2}(∑ r_{j}^{2}) ∏_{j=1}^{p} dr_{j}

= MW πϵ^{2} ∫_{D(1)} ∑_{j=1}^{p} r_{j}^{d_{j}+1} ∏_{i≠j} r_{i}^{d_{i}−1} ∏_{j=1}^{p} dr_{j}

We now make the change of variable x_{i} = r_{i}^{2} to evaluate each term in the sum. For each j, let b_{j} = d_{j}/2 and b_{i} = d_{i}/2 − 1, so that r_{j}^{d_{j}+1} dr_{j} = (1/2)x_{j}^{b_{j}} dx_{j} and r_{i}^{d_{i}−1} dr_{i} = (1/2)x_{i}^{b_{i}} dx_{i}. Then each term in the sum becomes

\frac{1}{2^{p}} ∫_{∑_{i} x_{i}≤1} x_{1}^{b_{1}} … x_{p}^{b_{p}} dx_{1} … dx_{p}

We have seen that this integral yields

\frac{1}{2^{p}} \frac{b_{1}! … b_{p}!}{(b_{1} + ⋯ + b_{p} + p)!}

Since b_{1} + ⋯ + b_{p} + p = ∑(d_{i}/2) + 1 = n/2, summing the terms gives

ν(C(H_{1}^{ϵ,th}(α, C)) = \frac{πϵ^{2}W M}{2^{p−1}} \frac{(∑(\frac{d_{i}}{2} )(\frac{d_{1}}{2} − 1)! … (\frac{d_{p}}{2} − 1)!}{(\frac{n}{2} − 1)(\frac{n}{2} − 2)!} =

= \frac{πϵ^{2}W M}{2^{p−1}} \frac{(\frac{d_{1}}{2} − 1)! … (\frac{d_{p}}{2} − 1)!}{(\frac{n}{2} − 2)!}

Hence,

c = lim_{ϵ→0} \frac{Vol(H_{1}^{ϵ,th}(α, C))}{πϵ^{2} Vol(H_{1}(α))} = \frac{M}{2^{p−1}(\frac{n}{2} − 2)! Vol(H_{1}(α))} ∏_{i=1}^{p}(\frac{d_{i}}{2} − 1)! Vol(H_{1}(α_{i}))

Thus, finally we get the following expression for the constant c in the setting where we fix the partition of named zeroes of S by components S′_{i} (see Problem 1 in section 7.4):

c = \frac{1}{|Γ|} · ∏_{j=1}^{p}(a_{j} + 1) · \frac{1}{2^{p−1}} · \frac{∏_{i=1}^{p}(\frac{d_{i}}{2} − 1)!}{(\frac{d}{2} − 2)!} · \frac{∏_{i=1}^{p} Vol(H_{1}(α_{i}))}{Vol(H_{1}(α))}

Let us study now the constant c in the setting where we do not specify which zeroes S are to be joined. (see Problem 2 in section 7.4). This counting problem can be reduced to the previous one by a purely combinatorial computation. Consider m ≠ m_{1}, m ≠ m_{2}, m ≠ a_{i}, i = 1, …, p. Then these zeroes are all “inherited” by the surfaces S_{i}. Since all the zeroes z_{1}, z_{2}, …, z_{l} are named, the number of ways to distribute o(m) zeroes of order m into groups of o_{1}(m), …, o_{p}(m) zeroes equals

\frac{o(m)!}{∏_{i=1}^{p} o_{i}(m)!}

If m ≠ m_{1} and m ≠ m_{2} but m = a_{j} for some j then one of the o_{j}(m) zeroes of order m which lives on S_{j} is newborn; that is, arising from the degeneration, while the other o_{j}(m) − 1 zeroes of order m come from the corresponding zeroes of order m on S. Thus the corresponding factor in the denominator becomes (o_{j}(m) − 1)!. Multiplying numerator and denominator by o_{j}(m) we get the following factor:

\frac{o(m)!}{∏_{j=1}^{p} o_{j}(m)!} · ∏_{j | a_{j}=m} o_{j}(m)

(Equivalently, we may say that we have performed the slit construction at the first “named” zero of order a_{j} on S_{j}. The rest of the zeroes of that order inherit their names on S, but then we count the number of ways of merging the names from different S_{i} onto S.) Consider now the case when m = m_{1} or m = m_{2}. When m_{1} ≠ m_{2} and m = m_{1} (resp. m = m_{2}) we have the arrangements of the other o(m_{1}) − 1 (resp. o(m_{2}) − 1) zeroes of order m. When m_{1} = m_{2} we have an arrangement of the other o(m_{1}) − 2 zeroes of order m. This changes the numerator in the formula above by the corresponding factorial. However, in these cases we have to count the number of ways to chose a zero of degree m_{1} and a zero of degree m_{2} on the initial surface S. This number is equal to o(m_{1}) · o(m_{2}) if m_{1} ≠ m_{2} and o(m_{1})(o(m_{1}) − 1) if m_{1} = m_{2}. Thus for zeroes of these degrees we have this extra factor o(m_{1}) · o(m_{2}) which combined with previous considerations gives o(m)! in this case as well. Taking into account the possible symmetries (see section 7.4), we finally get the following expression for the constant c in the setting where the zeroes of S are not numbered (see Problem 2 in section 7.4).

Formula 7.1.

(14) c = \frac{1}{|Γ_{−}|} · \frac{1}{|Γ|} · ∏_{m∈α} (\frac{o(m)!}{∏_{j=1}^{p} o_{j}(m)!}) · ∏_{\begin{matrix}j=1 \\ a_{j}≠0 \end{matrix}}^{p} o_{j}(a_{j})·

· ∏_{j=1}^{p}(a_{j} + 1) · \frac{1}{2^{p−1}} · \frac{∏_{i=1}^{p}(\frac{d_{i}}{2} − 1)!}{(\frac{d}{2} − 2)!} · \frac{∏_{i=1}^{p} Vol(H_{1}(α_{i}))}{Vol(H_{1}(α))}

We give some examples of this formula in the case of connected strata H_{1}(α).

Example 7.9. Stratum H_{1}(3, 1); The multiplicity is p = 2. After degeneration we get a surface S′_{1} of genus 2 with a single zero of order a_{1} = 2, where now a′_{1} = 2, a′′_{1} = 0, and we get a torus S′_{2}, with a_{2} = a′_{2} = a′′_{2} = 0.

The saddle connections γ_{1} and γ_{2} partition the cone angle 8π at z_{1} in two sectors with angles 2π and 6π; they partition the cone angle 4π at z_{2} in two sectors with angles 2π and 2π. There is no stratum interchange, so |Γ| = 1. We get

c = 3 · \frac{1}{2} · \frac{3! · 1!}{5!} \frac{Vol(H_{1}(2)) · Vol(H_{1}(∅))}{Vol(H_{1}(3, 1))} = \frac{567}{1024} ≈ 0.554

Example 7.10. Stratum H_{1}(2, 1, 1); genus g = 3, multiplicity p = 2. First suppose that we have a pair of homologous saddle connections joining the zero of degree m_{1} = 2 with one of two zeroes of degree m_{2} = 1. After degeneration we get two surfaces S′_{1} ∈ H_{1}(a_{1}, 1), and S′_{2}, ∈ H_{1}(a_{2}), where a_{1}+a_{2} = m_{1}+m_{2}+2−2p = 1. Thus a_{1} = 1, a_{2} = 0. There is no stratum interchange, so |Γ| = 1. Altogether we get

c = 4 · \frac{1}{2} · \frac{4! · 1!}{6!} · \frac{Vol(H_{1}(1, 1)) Vol(H_{1}(∅))}{Vol(H_{1}(2, 1, 1))} = \frac{28}{45} ≈ 0.622

Suppose now that we have a pair of homologous saddle connections joining two simple zeroes. After degeneration we get two surfaces S′_{1}, ∈ H(a_{1}, 2), and S′_{2}, ∈ H(a_{2}), where a_{1} + a_{2} = m_{1} + m_{1} + 2 − 2p = 0. Thus S′ is genus 2 with a marked point and with a zero of degree 2, while S′_{2} is a torus with a marked point. Here we have a′_{i} = a′′_{i} = 0, i = 1, 2.

There is no stratum interchange, so |Γ| = 1. We now get

c = \frac{1}{2} · 1 · \frac{4! · 1!}{6!} · \frac{Vol(H_{1}(2)) Vol(H_{1}(∅))}{Vol(H_{1}(2, 1, 1))} = \frac{7}{40} = 0.175

7.6. Principal stratum. The cone angle at any simple zero equals 4π. Thus for the principal stratum we may have at most two homologous saddle connections joining a pair of distinct zeroes. Hence in the higher multiplicity case, the surface S, ∈ H_{1}(1, …, 1) of genus g degenerates to a pair of surfaces S′_{1}, S′_{2} of genera g_{1} + g_{2} = g. Each of S′_{1}, S′_{2} has only simple zeroes and one additional marked point. We always have the γ → −γ symmetry, so |Γ_{−}| = 2. When g_{1} = g_{2} we have a symmetry |Γ| = 2 due to the stratum interchange; there is no other symmetry otherwise.

Applying formula (14) and (13) we finally get the following answer.

Formula 7.2. The constant for the number of saddle connections of multiplicity two joining two distinct zeroes in the principal stratum is

c = \frac{1}{2|Γ|} · \frac{(2g − 2)!}{(2g_{1} − 2)!(2g_{2} − 2)!} · 1 · 1 · \frac{1}{2^{2−1}} · \frac{(4g_{1} − 3)!(4g_{2} − 3)!}{(4g − 5)!}·

· \frac{Vol(H_{1}(\overbrace{1, …, 1}^{2g_{1}−2})) Vol(H_{1}(\overbrace{1, …, 1}^{2g_{2}−2}))}{Vol(H_{1}(\underbrace{1, …, 1}_{2g−2}))} =

= \frac{1}{4|Γ|} · \frac{(2g − 2)! (4g_{1} − 3)! (4g_{2} − 3)!}{(2g_{1} − 2)! (2g_{2} − 2)! (4g − 5)!} · \frac{Vol(H_{1}(\overbrace{1, …, 1}^{2g_{1}−2})) Vol(H_{1}(\overbrace{1, …, 1}^{2g_{2}−2}))}{Vol(H_{1}(\underbrace{1, …, 1}_{2g−2}))}

where g_{1} + g_{2} = g, g_{1}, g_{2} ≥ 1, |Γ| = \begin{cases}1 & when g_{1} ≠ g_{2} \\ 2 & when g_{1} = g_{2} \end{cases}

Figure 6. Saddle connection of multiplicity 2 on a surfaces from the principal stratum H_{1}(1, …, 1).

Table 4 presents the numerical values for these constants. More values are presented in Table 8 in the Appendix. (We present only approximate values of constants in Table 8 just to save the space.)

Table 4. Principal stratum; values of the constants for saddle connections of multiplicity two joining a pair of distinct zeroes

g_{2} = 1  g_{2} = 2  g_{2} = 3  g_{2} = 4  g_{2} = 5
g_{1} = 1  \frac{5}{8}  \frac{6}{7}  \frac{315}{377}  \frac{19604}{23357}  \frac{13897415}{16493303}
g_{1} = 2  \frac{6}{7}  \frac{30}{377}  \frac{1680}{23357}  \frac{686140}{16493303}  \frac{9529656}{351964697}
g_{1} = 3  \frac{315}{377}  \frac{1680}{23357}  \frac{154350}{16493303}  \frac{352872}{50280671}  \frac{389127620}{121780991823}
g_{1} = 4  \frac{19604}{23357}  \frac{686140}{16493303}  \frac{352872}{50280671}  \frac{336277214}{365342975469}  \frac{435911877856}{704782787198207}
g_{1} = 5 \frac{13897415}{16493303} \frac{9529656}{351964697} \frac{389127620}{121780991823} \frac{435911877856}{704782787198207} \frac{19865637635886}{249020069093788675}

8. Strata that are not Connected

The analysis of admissible assignments, structure of local surgeries, and most of the elements of our constructions are basically the same for the strata having several connected components. However, we need to establish the correspondence between connected components and parts of the boundary of the stratum they are adjacent to. In other words we need to describe the admissible constructions, which lead to the flat surfaces in the prescribe connected component.

8.1. Parity of the Spin Structure in the Slit Construction. Admissible Assignments for Surfaces from Hyperelliptic Components. We start with outlining the relation between the parities of spin structures of the resulting surface and its components in the slit construction.

Lemma 8.1. Suppose S constructed above has only even zeroes so has a spin structure. Then each component surface S′_{i} has only even zeroes and so also has a spin structure. The parity of the spin structure ϕ(S) is equal to the sum of the parities of the spin structures of the components:

(15)  ϕ(S) = ∑_{i=1}^{p} ϕ(S′_{i}) (mod 2)

Proof. For each S′_{i}, all the zeroes except the distinguished one come from the corresponding zeroes on S unchanged; in particular they have the same degrees as the initial zeroes of S. Thus for each S′_{i} all the zeroes but at most one, have even degrees. Note that the total sum of the degrees of the zeroes of S′_{i} is even: it equals 2g_{i} + 2, where g_{i} is the genus of the surface S′_{i}. Hence the remaining zero has even degree as well. (By usual convention we consider a marked point as a “zero of degree 0”.)

For each surface S′_{i} consider a collection of 2g_{i} smooth simple closed curves on S′_{i} representing the canonical basis of cycles. Breaking up the zero z_{i} we make a local surgery of the flat structure, i.e., we do not change the flat structure outside of a small domain U′_{i} ⊂ S′_{i} containing the zero z_{i}. Thus for each i we can deform if necessary the curves from the corresponding collection in such a way that they stay outside of a neighborhood of the zero z_{i}. Thus we may assume that for each i = 1, …, p none of the chosen curves on S′_{i} intersect U′_{i}. This means that all the curves survive under the surgery, moreover, the union of collections for all i gives us a canonical basis on S. By construction the index of any resulting curve in the flat structure ω is the same as the index of the corresponding curve in the flat structure ω′_{i}. Hence we get the desired relation (15). The lemma is proved. □

Lemma 8.2. Let S belong to one of the components H_{1}^{odd}(α) or H_{1}^{even}(α). Suppose that α ≠ (g − 1, g − 1). Then the surface S can be obtained by an assignment (S′, γ, a′_{i}, a′′_{i}) → (S, m_{1}, m_{2}, a′_{i}, a′′_{i}), where m_{1}, m_{2} ∈ α, if and only if the collection (α′_{i}, a′_{i}, a′′_{i}) satisfies the conditions of Lemma 7.1, all α′_{i} are even, and the parities of the spin structures of S′_{i} satisfy the condition of Lemma 8.1.

Note that the surfaces S′_{i} from the hyperelliptic component H_{1}^{hyp}(α) and from the hyperelliptic component H_{1}^{hyp}(g − 1, g − 1), where g is odd, also have parity of the spin structure (see formulae (7), (8)). In general these surfaces are also involved in assignments from Lemma 8.2 producing nonhyperelliptic components H_{1}^{even}(α) and H_{1}^{odd}(α). However, in the exceptional case of α = (g − 1, g − 1) with odd g some assignments lead to surfaces S from the hyperelliptic connected component H_{1}^{hyp}(g − 1, g − 1). These assignments are classified in the lemma below.

Lemma 8.3. Let a flat surface S from the hyperelliptic connected component H_{1}^{hyp}(g − 1, g − 1) be obtained by an assignment

(S′, γ, a′_{i}, a′′_{i}) → (S, g − 1, g − 1, a′_{i}, a′′_{i}).

Then for almost every S the assignment has one of the following two types:

— The multiplicity is 1; the flat surface S′ belongs to the hyperelliptic component H_{1}^{hyp}(α), and a′ = a′′ = g − 1.

— The multiplicity is 2; the flat surfaces S′_{i}, i = 1, 2 belong to the hyperelliptic components H^{hyp}(2g_{i} −2), where g_{1} +g_{2} = g, and a′_{i} = a′′_{i} = g_{i} −1, i = 1, 2.

The assignment is |Γ| to 1 and is onto the corresponding set of hyperelliptic surfaces.

Proof. The set of hyperelliptic structures such that the involution interchanges some components has positive codimension, so we may disregard these and assume the involution fixes each component. This implies that all flat surfaces S′_{i} belong to components H^{hyp}(2g_{i} − 2). Since the hyperelliptic involution sends a vector v to −v, but does not change the orientation of the surface, if the multiplicity is greater than one, it cannot map any of the γ′_{i} or γ′′_{i} to themselves. It must therefore interchange γ′_{i} and γ′′_{i} for each i. This implies that i = 2. □

Corollary 8.4. Almost any flat surface S from the hyperelliptic connected component H_{1}^{hyp}(g − 1, g − 1) has no saddle connections of multiplicity greater then two.

8.2. Constants for the Components of the Stratum H_{1}(g − 1, g − 1). We start with the stratum H_{1}(g −1, g −1) which is exceptional: it has the hyperelliptic connected component.

8.2.1. Hyperelliptic Connected Component H_{1}^{hyp}(g−1, g−1). The admissible assignments for the surfaces from the hyperelliptic connected component H_{1}^{hyp}(g−1, g−1) are described by Lemma 8.3. The possible multiplicities are one and two. In multiplicity one modifying (12) in accordance with Lemma 8.3 we get the following formula.

Formula 8.1. The constant for the number of saddle connections of multiplicity one joining distinct zeroes in H_{1}^{hyp}(g − 1, g − 1) is given by

c = (2g − 1) · \frac{Vol(H_{1}^{hyp}(2g − 2))}{Vol(H_{1}^{hyp}(g − 1, g − 1))}

In other words, collapsing a saddle connection of multiplicity one on a surface S ∈ H_{1}^{hyp}(g − 1, g − 1) we get a surface S′ from the hyperelliptic connected component H_{1}^{hyp}(α). Breaking up the zero of a surface S′ ∈ H_{1}^{hyp}(α) we necessarily get a surface S from the hyperelliptic component H_{1}^{hyp}(g − 1, g − 1).

Table 5. Hyperelliptic component H_{1}^{hyp}(g − 1, g − 1); values of the constants for saddle connections of multiplicity one joining distinct zeroes

g = 2 g = 3  g = 4  g = 5  g = 6  g = 7  g = 8  g = 9
c =  \frac{27}{8}  \frac{225}{32}  \frac{6125}{512}  \frac{297675}{16384} \frac{3361743}{131072} \frac{9018009}{262144} \frac{372683025}{8388608} \frac{59836330125}{1073741824}
c ≈ 3.375 7.03125 11.9629 18.1686 25.6481  34.401  44.4273  55.7269

Example 8.5. Component H_{1}^{hyp}(1, 1). In genus g = 2 the component H_{1}^{hyp}(1, 1) coincides with the whole principal stratum H_{1}(1, 1). The formula above gives the same constant as formula 6.3 for the principal stratum in genus g = 2:

c = 3 · \frac{Vol(H_{1}^{hyp}(2))}{Vol(H_{1}^{hyp}(1, 1))} = 3 · \frac{Vol(H_{1}(2))}{Vol(H_{1}(1, 1))} = \frac{27}{8} = 3.375

Example 8.6. Component H_{1}^{hyp}(2, 2). In genus 3 we have the connected component H_{1}^{hyp}(2, 2). After collapsing zeroes we obtain a hyperelliptic surface S′ ∈ H_{1}^{hyp}(4). We get

c = 5 · \frac{Vol(H_{1}^{hyp}(4))}{Vol(H_{1}^{hyp}(2, 2))} = \frac{225}{32} ≈ 7.03

We present more values in table 5.

Consider now multiple saddle connections for a flat surface S from the connected component H_{1}^{hyp}(g − 1, g − 1). By Lemma 8.3 we only have p = 2 in the higher multiplicity, and S′_{i} ∈ H^{hyp}(2g_{i} − 2), i = 1, 2, where g_{1} + g_{2} = g. Here we always have the γ → −γ symmetry, so |Γ_{−}| = 2. When g_{1} = g_{2} we also have the stratum interchange symmetry, so |Γ| = 2, and |Γ| = 1 otherwise. Thus, after appropriate modification of formula (14) we get the following constant

Formula 8.2. The constant in multiplicity 2 for saddle connections joining distinct zeroes in H_{1}^{hyp}(g − 1, g − 1) is given by

c = \frac{(2g_{1} − 1)(2g_{2} − 1)}{2|Γ|}·\frac{(2g_{1} − 1)!(2g_{2} − 1)!}{(2g − 1)!}·\frac{Vol(H_{1}^{hyp}(2g_{1} − 2))Vol(H_{1}^{hyp}(2g_{2} − 2))}{Vol(H_{1}^{hyp}(g − 1, g − 1))}

Example 8.7. Component H_{1}^{hyp}(1, 1). Saddle connections of multiplicity two joining distinct zeroes on a surface from H_{1}^{hyp}(1, 1) = H_{1}(1, 1) were actually, already considered in the section treating the principal stratum. Here after decomposition we get a pair of tori S′_{1}, S′_{2} each with a marked point. We get the following value for the corresponding constant:

c = \frac{1 · 1}{2 · 2} · \frac{1! · 1!}{3!} · \frac{Vol(H_{1}(∅)) Vol(H_{1}(∅))}{Vol(H_{1}(1, 1))} = \frac{5}{8} = 0.625

Table 6. Hyperelliptic component H_{1}^{hyp}(g − 1, g − 1); values of the constants for saddle connections of multiplicity two joining distinct zeroes

g_{2} = 1  g_{2} = 2  g_{2} = 3  g_{2} = 4  g_{2} = 5  g_{2} = 6
g_{1} = 1  \frac{5}{8}  \frac{63}{32}  \frac{375}{128}  \frac{67375}{16384}  \frac{361179}{65536}  \frac{1867635}{262144}
g_{1} = 2  \frac{63}{32}  \frac{567}{512}  \frac{22275}{8192}  \frac{441441}{131072}  \frac{1083537}{262144}  \frac{10496871}{2097152}
g_{1} = 3  \frac{375}{128}  \frac{22275}{8192}  \frac{96525}{65536}  \frac{875875}{262144}  \frac{4027725}{1048576}  \frac{1187146125}{268435456}
g_{1} = 4  \frac{67375}{16384}  \frac{441441}{131072}  \frac{875875}{262144}  \frac{14889875}{8388608}  \frac{2083228875}{536870912}  \frac{18432519875}{4294967296}
g_{1} = 5 \frac{361179}{65536}  \frac{1083537}{262144}  \frac{4027725}{1048576}  \frac{2083228875}{536870912}  \frac{8749561275}{4294967296}  \frac{74923166241}{17179869184}
g_{1} = 6 \frac{1867635}{262144} \frac{10496871}{2097152} \frac{1187146125}{268435456} \frac{18432519875}{4294967296} \frac{74923166241}{17179869184} \frac{624359718675}{274877906944}

Example 8.8. Component H_{1}^{hyp}(2, 2). Consider saddle connections of multiplicity two joining distinct zeroes on a surface from H_{1}^{hyp}(2, 2). After decomposition we get a surface S′_{1} ∈ H_{1}(2) of genus 2 with a single zero of order 2 and a torus S′_{2} ∈ H(∅) with a marked point. The cone angle between γ_{1} and γ_{2} is the same at z_{1} and z_{2}; it equals 2π. The cone angle between γ_{2} and γ_{1} is also the same at z_{1} and z_{2}, but now it equals 4π. The constant is given by

c = \frac{3 · 1}{2 · 1} · \frac{3! · 1!}{5!} · \frac{Vol(H_{1}(2)) Vol(H_{1}(∅))}{Vol(H_{1}^{hyp}(2, 2))} = \frac{63}{32} ≈ 1.97

We present more values in table 6 and in table 9 in the Appendix.

8.2.2. Connected Component H_{1}^{nonhyp}(g −1, g −1); Even Genus. For g ≥ 4, g even, the stratum H_{1}(g−1, g−1) has exactly two connected components: H_{1}^{hyp}(g−1, g−1) and H_{1}^{nonhyp}(g−1, g−1). Thus, dealing with the connected component H_{1}^{nonhyp}(g− 1, g −1) we have to exclude all the assignments producing surfaces S in the stratum H_{1}(g − 1, g − 1) those that actually belong to the component H_{1}^{hyp}(g − 1, g − 1). The latter problem has already been solved. Thus we get the following answers.

Formula 8.3. For any even genus g ≥ 4 any flat surfaces S from the connected component H_{1}^{nonhyp}(g − 1, g − 1) having a saddle connection of multiplicity one joining distinct zeroes may be obtained from the corresponding surface S′ from one of the components H_{1}^{even}(2g − 2), H_{1}^{odd}(2g − 2).

The corresponding constant is equal to

c = (2g − 1) · \frac{Vol(H_{1}^{even}(2g − 2)) + Vol(H_{1}^{odd}(2g − 2))}{Vol(H_{1}^{nonhyp}(g − 1, g − 1))} =

= (2g − 1) · \frac{Vol(H_{1}(2g − 2)) − Vol(H_{1}^{hyp}(2g − 2))}{Vol(H_{1}(g − 1, g − 1)) − Vol(H_{1}^{hyp}(g − 1, g − 1))}

We proceed with multiplicity two. Here we get the formula analogous to formula 8.2 for the component H_{1}^{hyp}(g − 1, g − 1).

Formula 8.4. The constant for multiplicity 2 joining two zeroes in the stratum H^{nonhyp}(g − 1, g − 1) is given by

c = \frac{(2g_{1} − 1)(2g_{2} − 1)}{2|Γ|} · \frac{(2g_{1} − 1)!(2g_{2} − 1)!}{(2g − 1)!}·

· \frac{Vol(H_{1}(2g_{1} − 2)) Vol(H_{1}(2g_{2} − 2)) − Vol(H_{1}^{hyp}(2g_{1} − 2)) Vol(H_{1}^{hyp}(2g_{2} − 2))}{Vol(H_{1}^{nonhyp}(g − 1, g − 1))},

For multiplicity p ≥ 3 it is easy to account for the impact of the hyperelliptic component (see Corollary 8.4). We get the following particular case of formula 14 in the case of H^{nonhyp}(g − 1, g − 1)

Formula 8.5.

c = \frac{1}{2|Γ|} · ∏_{i=1}^{p}(2g_{i} − 1) · \frac{∏_{i=1}^{p}(2g_{i} − 1)!}{(2g − 1)!} · \frac{∏_{i=1}^{p} Vol(H_{1}(2g_{i} − 2))}{Vol(H_{1}^{nonhyp}(g − 1, g − 1))}

8.2.3. Components H^{even}(g − 1, g − 1) and H^{odd}(g − 1, g − 1). We consider the multiplicity one case first. Assume that S belongs to one of the nonhyperelliptic components H^{even}(g−1, g−1) or H^{odd}(g−1, g−1). Contracting the saddle connection joining the two zeroes, we merge the two zeroes into one and we get a surface S′ in H^{even}(2g − 2) or H^{odd}(2g − 2) correspondingly. Conversely breaking up a single zero of a surface S′ from a nonhyperelliptic connected component H^{even}(2g − 2) or H^{odd}(2g − 2) into two zeroes of degrees g − 1 we get a surface from the component H^{even}(g − 1, g − 1) or H^{odd}(g − 1, g − 1) correspondingly. Thus we have to modify formula (12) in the following way.

Formula 8.6. The constant is equal to

c = (2g − 1) · \frac{Vol(H_{1}^{even}(2g − 2))}{Vol(H_{1}^{even}(g − 1, g − 1))}

c = (2g − 1) · \frac{Vol(H_{1}^{odd}(2g − 2))}{Vol(H_{1}^{odd}(g − 1, g − 1))}

Example 8.9. Component H^{odd}(2, 2). After collapsing zeroes we obtain a surface S′ ∈ H^{odd}(4). We get

c = 5 · \frac{Vol(H_{1}^{odd}(4))}{Vol(H_{1}^{odd}(2, 2))} = \frac{80}{9} ≈ 8.89

Remark 8.10. Note that the similar nonhyperelliptic connected components with even parity of the spins structure do not exist in genus g = 3.

Now we consider higher multiplicity. We again have to consider multiplicity two separately because some assignments in multiplicity two produce surfaces from the hyperelliptic connected component H^{hyp}(g − 1, g − 1), see Lemma 8.1. We again get the formula analogous to formula 8.2 for the component H^{hyp}(g − 1, g − 1) and to formula 8.4 for the component H^{nonhyp}(g − 1, g − 1).

It will be convenient to introduce the following notation. We introduce a function δ(α, ϕ) which has values 0 or 1. It is equal to 1 when all of the following three conditions are satisfied: all the zeroes of α (if any) have even degrees; H(α) contains the hyperelliptic component H^{hyp}(α); the parity of the spin structure of H^{hyp}(α) coincides with ϕ. Otherwise δ(α, ϕ) = 0.

Formula (7) for the parity of the spin structure of H^{hyp}(α′) shows that

(16)  δ(α, ϕ) = \begin{cases}1 & when α = (2g − 2) and [\frac{g + 1}{2}] = ϕ (mod 2) \\ 1 & when α = (g − 1, g − 1), g is odd, \\ & and (\frac{g + 1}{2}) = ϕ (mod 2) \\ 0 & otherwise \end{cases}

We shall also use the following convention: the volume of a nonexistent component is equal to zero. In this notation we get the following six “dummy” volumes for the strata in small genera:

Vol(H_{1}^{odd}(∅)) = 0  Vol(H_{1}^{odd}(2)) = 0  Vol(H_{1}^{even}(4)) = 0
(17)  Vol(H_{1}^{even}(∅)) = 0  Vol(H_{1}^{even}(2)) = 0  Vol(H_{1}^{even}(2, 2))= 0

and also we have

H^{hyp}(α) = 0 if α ≠ (2g − 2), (g − 1, g − 1)

Formula 8.7. A surface S from one of the nonhyperelliptic connected components H^{even}(g−1, g−1) or H^{odd}(g−1, g−1) may have a saddle connection of multiplicity two joining the two distinct zeroes if and only if it can be obtained from surfaces S′_{1} and S′_{2} of genus g_{1}, g_{2} where g_{1} + g_{2} = g and where the following additional requirements are satisfied. At least one of the surfaces S′_{1}, S′_{2} does not belong to the hyperelliptic component H^{hyp}(2g_{i} −2), i = 1, 2; the parities of the spin structures of the surfaces satisfy relation (15): ϕ(S′_{1}) + ϕ(S′_{2}) = ϕ(S). For S ∈ H^{odd}(g − 1, g − 1) the spin structures of S′_{1}, S′_{2} should have opposite parities. This means in particular that there is no longer a stratum interchange symmetry. For g_{1} ≠ g_{2}

c = \frac{(2g_{1} − 1)(2g_{2} − 1)}{2} · \frac{(2g_{1} − 1)!(2g_{2} − 1)!}{(2g − 1)!} · \frac{1}{Vol(H_{1}^{odd}(g − 1, g − 1))}·

((( Vol(H_{1}^{even}(2g_{1} − 2)) + δ((2g_{1} − 2), even) Vol(H_{1}^{hyp}(2g_{1} − 2)))·

· ( Vol(H_{1}^{odd}(2g_{2} − 2)) + δ((2g_{2} − 2), odd) Vol(H_{1}^{hyp}(2g_{2} − 2))))+

+ (( Vol(H_{1}^{odd}(2g_{1} − 2)) + δ((2g_{1} − 2), odd) Vol(H_{1}^{hyp}(2g_{1} − 2)))·

· ( Vol(H_{1}^{even}(2g_{2} − 2)) + δ((2g_{2} − 2), even) Vol(H_{1}^{hyp}(2g_{2} − 2))))−

− δ((2g_{1} − 2), even)δ((2g_{2} − 2), odd) Vol(H_{1}^{hyp}(2g_{1} − 2)) Vol(H_{1}^{hyp}(2g_{2} − 2))−

− δ((2g_{1} − 2), odd)δ((2g_{2} − 2), even) Vol(H_{1}^{hyp}(2g_{1} − 2)) Vol(H_{1}^{hyp}(2g_{2} − 2)))

For g_{1} = g_{2} = g/2, where g is even, we get

c = \frac{(2g_{1} − 1)^{2}}{2} · \frac{((2g_{1} − 1)!)^{2}}{(2g − 1)!} · \frac{1}{Vol(H_{1}^{odd}(g − 1, g − 1))}·

· ( Vol(H_{1}^{even}(2g_{1} − 2)) + δ((2g_{1} − 2), even) Vol(H_{1}^{hyp}(2g_{1} − 2)))·

· ( Vol(H_{1}^{odd}(2g_{2} − 2)) + δ((2g_{2} − 2), odd) Vol(H_{1}^{hyp}(2g_{2} − 2)))

Example 8.11. Component H^{odd}(2, 2). From Formula 8.7 we know that the degeneration of multiplicity 2 of a surface from H(2, 2) has the form (\overline{0} + \overline{0}) ≻ (\overline{1} + \overline{1}) ≻. Since the strata H(∅) and H(2) are (hyper)elliptic the resulting surface in H(2, 2) is also hyperelliptic, see Lemma 8.1. Hence degeneration of multiplicity 2 is not realizable for flat surfaces from H^{odd}(2, 2).

The expression for the constant for the surfaces from H^{even}(g − 1, g − 1) is analogous to the one above with the only exception that now we get the stratum interchange symmetry |Γ| = 2 for g_{1} = g_{2}:

c = \frac{(2g_{1} − 1)(2g_{2} − 1)}{2|Γ|} · \frac{(2g_{1} − 1)!(2g_{2} − 1)!}{(2g − 1)!} · \frac{1}{Vol(H_{1}^{even}(g − 1, g − 1))}·

((( Vol(H_{1}^{even}(2g_{1} − 2)) + δ((2g_{1} − 2), even) Vol(H_{1}^{hyp}(2g_{1} − 2)))·

· ( Vol(H_{1}^{even}(2g_{2} − 2)) + δ((2g_{2} − 2), even) Vol(H_{1}^{hyp}(2g_{2} − 2))))+

+ (( Vol(H_{1}^{odd}(2g_{1} − 2)) + δ((2g_{1} − 2), odd) Vol(H_{1}^{hyp}(2g_{1} − 2)))·

· ( Vol(H_{1}^{odd}(2g_{2} − 2)) + δ((2g_{2} − 2), odd) Vol(H_{1}^{hyp}(2g_{2} − 2))))−

− δ((2g_{1} − 2), even)δ((2g_{2} − 2), even) Vol(H_{1}^{hyp}(2g_{1} − 2)) Vol(H_{1}^{hyp}(2g_{2} − 2))−

− δ((2g_{1} − 2), odd)δ((2g_{2} − 2), odd) Vol(H_{1}^{hyp}(2g_{1} − 2)) Vol(H_{1}^{hyp}(2g_{2} − 2)))

Finally, for from multiplicity at least three it is easy to account for the impact of the hyperelliptic component (see Corollary 8.4). We get the following particular case of formula 14:

Formula 8.8. Almost every surface in H^{even}(g − 1, g − 1) or in H^{odd}(g − 1, g − 1) having saddle connections of multiplicity p ≥ 3 joining two zeroes can be obtained by assignment

(\overline{(g_{1} − 1)}) + \overline{(g_{1} − 1)}) ≻ … (\overline{(g_{p} − 1)} + \overline{(g_{p} − 1)}) ≻

where g_{i} ≥ 1, i = 1, …, p, and g_{1} + ⋯ + g_{p} = g − 1. The corresponding constant equals

c = \frac{1}{|Γ|} · ∏_{i=1}^{p}(2g_{i} − 1) · \frac{1}{2^{p−1}} · \frac{∏_{i=1}^{p}(2g_{i} − 1)!}{(2g − 1)!} · \frac{1}{Vol(H_{1}^{ϕ}(g − 1, g − 1))}·

· ∑_{\begin{matrix}ϕ_{1},...,ϕ_{p}∈\{even,odd\} \\ ϕ_{1}+···+ϕ_{p}=ϕ (mod 2) \end{matrix}} ∏_{i=1}^{p} ( Vol(H_{1}^{ϕ_{i}}(2g_{i} − 2)) + δ((2g_{i} − 2), ϕ_{i}) Vol(H_{1}^{hyp}(2g_{i} − 2)))

Note that here we always have the γ → −γ symmetry, |Γ_{−}| = 2, which is cancelled by o(m)! = o(g − 1)! (see (14)). Note that in this particular case we prefer to use summation over all combinations of parities ϕ_{i} producing the proper parity of the sum. Thus the stratum interchange symmetry Γ depends only on the collection of g_{i}, and on their cyclic order.

Example 8.12. Component H^{odd}(2, 2). Consider a saddle connection joining the two zeroes of a surface S ∈ H^{odd}(2, 2), and suppose that the saddle connection has multiplicity 3. Each of S′_{i}, for i = 1, 2, 3 is a torus with a marked point. The angle between any consecutive saddle connections at z_{1} and z_{2} equals 2π. There is a symmetry of order 3 coming from stratum interchange, so |Γ| = 3. Taking into consideration 1/4 from 1/2^{p−1} we get

c = \frac{1}{3} · \frac{1}{4} · \frac{1! · 1! · 1!}{5!} · \frac{(Vol(H_{1}(∅)))^{3}}{Vol(H_{1}^{odd}(2, 2))} = \frac{1}{9} ≈ 0.111

Remark 8.13. Let g ≥ 4 be odd. We see that the maximal multiplicity of a saddle connection joining the distinct zeroes of a surface from the stratum H(g − 1, g − 1) is different for different connected components: it equals 2 for almost all surfaces from H^{hyp}(g − 1, g − 1), it equals g − 1 for almost all surfaces in H^{even}(g − 1, g − 1); it equals g for almost all surfaces in H^{odd}(g − 1, g − 1).

8.3. Other Strata. The description for all other strata H(α), where all the entries of α are even, is similar to the one just presented.

Formula 8.9. The constant for the nonhyperelliptic connected component H^{ϕ}(α), ϕ ∈ {even, odd}, α ≠ (g − 1, g − 1), is given by

c = \frac{1}{|Γ_{−}|} · \frac{1}{|Γ|} · ∏_{m∈α} (\frac{o(m)!}{∏_{j=1}^{p} o_{j}(m)!})· ∏_{\begin{matrix}j=1 \\ a_{j}≠0 \end{matrix}}^{p} o_{j}(a_{j})· ∏_{j=1}^{p}(a_{j} +1)· \frac{1}{2^{p−1}} · \frac{∏_{i=1}^{p}(\frac{d_{i}}{2} − 1)!}{(\frac{d}{2} − 2)!}·

· \frac{1}{Vol(H_{1}^{ϕ}(α))} · ∑_{\begin{matrix}ϕ′_{1},...,ϕ′_{p}∈\{even, odd\} \\ ϕ′_{1}+···+ϕ′_{p}=ϕ (mod 2) \end{matrix}} ∏_{i=1}^{p} ( Vol(H_{1}^{ϕ_{i}}(α′_{i})) + δ(α′_{i}, ϕ′_{i}) Vol(H_{1}^{hyp}(α′_{i})))

Example 8.14. Stratum H^{even}(4, 2). In multiplicity one, after collapsing zeroes we obtain a surface S′ ∈ H(6), where either of the two components H^{hyp}(6) and H^{even}(6) having even parity of the spin structure are possible. We get

c = 7 · \frac{(Vol(H_{1}^{hyp}(6)) + Vol(H_{1}^{even}(6)))}{Vol(H_{1}^{even}(4, 2))} = \frac{253001}{18225} ≈ 13.88

In multiplicity two we get two surfaces S′_{1} ∈ H(α′_{1}), S′_{2} ∈ H(α′_{2}), where α′_{1}+α′_{2} = m_{1} + m_{2} − p = 4.

There are two possibilities. In the first case α′_{1} = α′_{2}. Note that any S′ ∈ H(2) has odd parity of the spin structure, see (7). Thus the parity of the resulting flat structure S is even, see Lemma 8.1.

The saddle connections γ_{1} and γ_{2} partition the cone angle 10π at z_{1} in two sectors with angles 8π and 2π; they partition the cone angle 6π at z_{2} in two sectors with the angles 4π and 2π. Here a′_{1} = 2, a′′_{1} = 0 while a′_{2} = a′′_{2} = 1 so that while H(α′_{1}) = H(α′_{2}) = H(2), there is no stratum interchange since a′_{1} ≠ a′_{2}. Thus |Γ| = 1. We get

c = 9 · \frac{1}{2} · \frac{3! · 3!}{7!} · \frac{(Vol(H_{1}(2)))^{2}}{Vol(H_{1}^{even}(4, 2))} = \frac{6}{25} = 0.24

In the second case after degeneration we get S′_{1} ∈ H_{1}(∅), S′_{2} ∈ H_{1}(4) Since H(∅) has odd parity of the spin structure, S′_{2} ∈ H^{odd}(4) in order to result in a surface S ∈ H^{even}(4, 2) (see Lemma 8.1).

The saddle connections γ_{1} and γ_{2} partition the cone angle 10π at z_{1} in two sectors with the angles 2π and 8π; they partition the cone angle 6π at z_{2} in two sectors with the angles 2π and 4π. We get

c = 5 · \frac{1}{2} · \frac{1! · 5!}{7!} · \frac{Vol(H_{1}(∅)) Vol(H_{1}^{odd}(4))}{Vol(H_{1}^{even}(4, 2))} = \frac{640}{729} ≈ 0.8779

Multiplicity 3 is not realizable in the component H^{even}(4) (see the next subsection).

Example 8.15. Stratum H^{odd}(4, 2). This component is similar in many aspects to the previous one, so we skip those details of calculations which are common for both components.

The first case is H^{odd}(4, 2) with multiplicity p = 1. After collapsing zeroes we obtain a surface S′ ∈ H(6), the only component that results is H^{odd}(6). We get

c = 7 · \frac{(Vol(H_{1}^{odd}(6))}{Vol(H_{1}^{odd}(4, 2))} = \frac{147}{10} = 14.7

Consider multiplicity 2. The case of degeneration to two surfaces S′_{1}, S′_{2} ∈ H(2) does not take place here, since the surface S ∈ H(4, 2) obtained from two surfaces from H(2) has even parity of spin structure.

Thus, the only possibility is S′_{1} ∈ H_{1}(∅); S′_{2} ∈ H_{1}(4), a′_{2} = 3, a′′_{2} = 1. Since H(∅) has odd parity of the spin structure, S′_{2} must be in components of H(4) having even parity of spin structure in order to result a surface S ∈ H^{odd}(4, 2) (see Lemma 8.1). There is only one such component: H^{hyp}(4).

Thus we get

c = 5 · \frac{1}{2} · \frac{1! · 5!}{7!} · \frac{Vol(H_{1}(∅)) Vol(H_{1}^{hyp}(4))}{Vol(H_{1}^{odd}(4, 2))} = \frac{15}{64} ≈ 0.2344

(see analogous calculation for the component H^{even}(4, 2).

Consider now multiplicity 3. It is not hard to check that S′_{1} ∈ H(2), S′_{2}, S′_{3} ∈ H(∅). Note that both H(∅) and H(2) have odd parity of the spin structure. Thus the resulting surface S ∈ H(4, 2) always has odd parity of the spin structure, see Lemma 8.1.

The saddle connections γ_{1}, γ_{2}, γ_{3} partition the cone angle 10π at z_{1} in three sectors with the angles 6π, 2π, and 2π; they partition the cone angle 6π at z_{2} in three sectors with the angles 2π. We find

c = 3 · \frac{1}{4} · \frac{3! · 1! · 1!}{7!} · \frac{Vol(H(2))(Vol(H_{1}(∅)))^{2}}{Vol(H_{1}^{odd}(4, 2))} = \frac{21}{320} ≈ 0.06563

Part 2. Saddle Connections Joining a Zero to Itself

9. Principal Boundary

In this part we consider the second problem of this paper. As described in Section 3, we have a surface S and a saddle connection γ_{1} joining a zero to itself. There may be other saddle connections γ_{i} which are homologous to γ_{1}. Suppose the curves γ_{i} bound q cylinders. The complement of the curves and cylinders are p = m−q surfaces S_{1}, …, S_{p}. The sum of the genera of the S_{i} is g−1. The surfaces S_{i} are attached in some cyclic order. Thus for any i the “next surface” S_{i+1} and the previous surface S_{i−1} make sense mod p.

We associate a set J = {i_{1}, …, i_{q}} ⊂ {1, …, p}, where 0 ≤ q ≤ p with the property that if i = i_{j} ∈ J then there is a cylinder between γ_{i_{j}} and γ_{i_{j}+1}. Now suppose a surface S_{i} is bounded by two homologous curves that leave and return to the same zero. The two curves bound angles 2(a′_{i} + 1)π and 2(a′′_{i} + 1)π at the zero. The two boundary circles are joined together at the zero to form a “figure eight”. If the two homologous curves that bound S_{i} return to different zeroes, then they bound angles b′_{i} and b′′_{i} at those zeroes. The boundary circles are disjoint in that case.

Suppose S_{i} is a surface whose boundary components are joined at a point to form a figure eight; that is we may think of that figure eight as a pair of circles with an identification of a point on one with a point on another. We may separate the boundary circles at the common point, by removing the identification. This gives Figure 7. Homologous closed geodesics of high multiplicity. Topological picture.

a surface with a single circular boundary component with two singular points on it with angles 2π(a′_{i} + 1) and 2π(a′′_{i} + 1). The boundary component is made up of a pair of saddle connections of equal length joining the two zeroes. Now we identify the two saddles, giving a closed surface with a saddle connection joining the two zeroes. We then collapse the saddle into a zero of order a_{i} = a′_{i} + a′′_{i}.

Now let S_{i} be a surface whose boundary circles are disjoint. There are 2b′_{i} + 1 and 2b′′_{i} + 1 sectors at the cone points that are not adjacent to the boundary circles which are cyclically ordered. For one circle the first of these is outgoing and for the other incoming. We shall want to “open up” the circles. The construction is not quite canonical and depends on the notion of transversal, which we now discuss.

Definition 9.1. Given a pair of points z_{0}, z_{1} on a surface and a vector γ, a transversal joining z_{0}, z_{1} is a smooth path ρ(t), 0 ≤ t ≤ 1, ρ(0) = z_{0}, ρ(1) = z_{1}, such that for all 0 < t < 1, ρ(t) is a regular point and such that the scalar product of the tangent vector to ρ with γ is strictly positive along the path, ⟨ρ̇, γ⟩ > 0 for any t.

Notice that given a zero of order k and direction γ there are k + 1 outgoing sectors at the zero which means that we can connect the zero to a point in the sector by a transversal in the sector. Similarly, there are k + 1 incoming sectors which means that we can connect a point in the sector to the zero by a transversal in the sector. We need to show that transversals exist joining two zeroes and an outgoing sector at one with an incoming sector at the other. This follows from the Calabi Theorem [C], or from the results in [HM]. However, for the sake of completeness we give a proof here.

Proposition 9.2. Given a flat surface S a direction γ, a pair of zeroes with an outgoing sector for the first zero and an incoming sector for the second, then there is a transversal joining the two zeroes in those sectors.

Proof. Suppose ω defines the flat surface. We may define a holomorphic 1-form ω̃ = e^{iϕ}ω such that Im(ω̃) vanishes along the directions γ and Re(ω̃) is positive along the direction γ. Let z_{0} and z_{1} any two regular points. Let U = U(z_{0}) consist of the set of points such that there is a transversal ρ joining z_{0} to z and such that Im(ω̃) > 0 along ρ. Then U is an open set. We show that it is the entire surface punctured at the zeroes, so that it must contain z_{1}.

It is easy to see that if the boundary ∂U contains a regular point w ∈ S, it contains the trajectory passing in direction γ through w. Let γ^{⊥} be the direction orthogonal to γ, and such that Im(ω̃) is positive on γ^{⊥}. Let ∂U = ∂U^{+} ⊔ ∂U^{−}, where γ^{⊥} is the inner normal to ∂U^{−}, where γ^{⊥} is the inner normal to ∂U^{−}, and outer normal to ∂U^{+}. It is easy to see that ∂U^{+} contains only the isolated points — the zeroes of ω. Thus the 1-cycle represented by ∂U^{−} is homologous to zero. Hence, since Re(ω̃) is a closed 1-form we have ∫_{∂U^{−}} Re(ω̃) = 0. On the other hand, if ∂U^{−} contains some curves, then they all have direction γ and Re(ω̃) is positive in direction γ and we have a contradiction. Thus ∂U^{−} contains only zeroes of ω.

Now given a pair of zeroes z and w, choose z_{0} in the outgoing sector at z and w_{0} in the incoming sector at w. Join z_{0} to w_{0} by a transversal. Connect up z to z_{0} by a transversal and w_{0} to w by a transversal. The juxtaposition of these paths is the required transversal joining z to w. □

Choose τ_{i} to be a transversal joining the cone points in these two sectors. Before proceeding further we also prove the following lemma which says that we may take τ_{i} to have bounded length.

Lemma 9.3. There exists M′′ such that for all S ∈ C(H_{1}^{ϵ}(α)) and any two zeroes z_{0}, z_{1} in the same component S_{i} of the complement of the γ_{j}, there is a transversal τ_{i} of length at most M′′ joining z_{0} to z_{1} in that component. Moreover there is a constant c such that |τ_{i} · β| ≤ c for any element β of the homology basis

Proof. Suppose the result is false. Then there exists a sequence S_{n} of flat structures, a sequence of directions γ_{n}, and zeroes z_{0}, z_{1} lying in the same components of the complement of the curves and cylinders on S_{n} such that the shortest transversals joining z_{0} to z_{1} in the component of the complement has length M_{n} → ∞. We can normalize so that S_{n} has unit area. By passing to subsequences we can assume that there is a surface S_{∞}, a set of zeroes and distinguished points of S_{∞} such that S_{n} converges to S_{∞} and the convergence is uniform from the complement of annular neighborhoods V_{n} of C_{i} and the β_{i} to the complement of a neighborhood U of the marked points and zeroes of S_{∞}. We can assume z_{0}, z_{1} correspond to zeroes or marked points w_{0} and w_{1} of S_{∞} and the directions γ_{n} converge to a direction γ. By Proposition 9.2 we join w_{0} to w_{1} by a transverse path ρ. Let w′_{0}, w′_{1} be the points where ρ leaves and then returns to U. Call this subsegment ρ′. By uniform convergence there is a transversal ρ_{n} on S_{n} for large n close to ρ′, leaving and returning to V_{n} at points z′_{0} and z′_{1}. We now can connect up z′_{0} and z′_{1} to z_{0} and z_{1} inside V_{n} to find a bounded length transversal.

Since the lengths of the transversals τ_{i} are bounded by M′′, the algebraic intersections β · τ_{i} are bounded by some constant c: if not, there would be sequences of flat structures S with transversals of length at most M such that the algebraic intersection goes to infinity. Passing to subsequences we can assume the flat structures converge and the transversal converge, and we have a contradiction. □

We continue with the discussion of the principal boundary. For S_{i} whose boundary components consists of a pair of circles, cut S_{i} along a transversal τ_{i} of length at most M′′. This gives a surface minus a parallelogram with two sides parallel to τ_{i} and two sides parallel to the boundary circles. Fill in the parallelogram to gives a closed surface S′_{i}. We note that the resulting S′_{i} depends on choice of τ_{i}. Altogether, using the figure eight and parallelogram degenerations the original surface S (of genus g) degenerates into p = m − q disjoint surfaces S′_{1}, …, S′_{p}. The sum of the genera of the S′_{i} is g − 1. The surfaces S′_{i} are attached in the same cyclic order as the S_{i}. The resulting surfaces belong to the principal boundary corresponding to the configuration.

Finally we note that the holonomy for S and S′_{i} of a basis of curves β of the S′_{i} satisfies

v_{S}(β) = v_{S′_{i}}(β) + (∑_{i} β · τ_{i})γ

10. Constructing surfaces with homologous saddle connections

Our objective now is to reverse the degeneration and build homologous saddle connections with associated configuration (J, a′_{i}, a′′_{i}, b′_{k}, b′′_{k}) out of simpler surfaces. We shall start with a collection of closed surfaces and via a “figure eight” and “parallelogram construction” build a closed surface with a curve or set of homologous curves joining a zero to itself.

2π(a′ + 1)

Figure 8. Here we applied the figure eight construction with a′ = 0 and a′′ = 4 to a zero of order a = a′ + a′′ = 4, and then we glued in a cylinder.

10.1. Figure Eight Construction. Given a surface S′_{i}, a zero (or a marked point) z_{i} of order a_{i}, and a pair of numbers a′_{i} + a′′_{i} = a_{i} we may break up the zero and perform the slit construction. We may then identify the two endpoints of the slit. This gives a surface with two circular boundary components γ′_{i}, γ′′_{i}, which are joined at a point. The two circular boundary components separate the total angle at the point into angles 2π(a′_{i} + 1) and 2π(a′′_{i} + 1). By convention on the choice of γ′_{i}, γ′′_{i} (see section 7.2) turning around z_{i} in a clockwise direction from γ′′_{i} to γ′_{i} inside the surface S′_{i} we turn by the angle 2π(a′_{i} + 1).

Recall that we may break the zero of order a_{i} up in a_{i} + 1 ways. Thus the figure eight construction, actually, gives us a_{i} + 1 surfaces.

In the second construction, given a pair of distinct points and a vector, we shall “open” the surface at the two points using the vector to build a surface with two boundary components. Again the construction is not quite canonical and depends on choice of transversal.

10.2. Parallelogram Construction. Suppose z and w are zeroes of orders b′, b′′ ≥ 0 on a surface S′, γ is a vector of length ϵ. Given separatrices leaving z and w in direction γ the notion of z′ as the point z + γ and w′ as the point w + γ make sense for small γ. At the point z there are two sectors adjacent to the segment from z to z′. Choose the outgoing sector. Similarly choose the incoming sector at w adjacent to the segment from w to w′. Let ρ be a transversal joining z to w transversal to γ in those sectors. For |γ| small enough parallel translations of ρ by the vectors sγ, 0 < s ≤ 1 are well defined, and the translated curves are adjacent to the segments from z to z′ and w to w′ Denote by ρ′ the parallel translation of ρ by γ. Now, cut S′ along the segments \overline{zz′}, \overline{ww′}, and along the curves ρ, ρ′. Remove the interior of a resulting “parallelogram” with vertices z, z′, w, w′. Now identify the two sides of the “parallelogram” parallel to ρ. This identifies z with z′ and w with w′ and creates a surface with two circular boundary pieces which we denote \overline{zz′} and \overline{ww′}. We note that the resulting flat structure depends on the choice of the separatrices leaving each zero, the choice of transversal as well as the other given data.

a_{1} a_{1} b_{1}  b_{1}  a_{2}  a_{2} b_{2}  b_{2} c_{1} c_{1} c_{2} c_{2} d_{1} d_{1}  d_{2} d_{2} γ⃗  γ⃗

Figure 9. In this example we apply the parallelogram construction simultaneously to two tori (represented by large parallelograms with identified opposite sides) and then glue in a pair of cylinders (represented by the narrow stripes with identified opposite sides).

Example 10.1. In the example presented at Figure 9 we take two flat tori. On the picture they are represented by the large parallelograms with identified opposite sides a_{1} to a_{1}, b_{1} to b_{1}, and a_{2} to a_{2}, b_{2} to b_{2}. In fact, zeroes z_{1}, w_{1}, and z_{2}, w_{2} are fake here: all these points are just the marked points of the tori. In this example we chose vector γ⃗ to be vertical.

We cut a rectangular domain out of the interior of each of the two tori in such way that the vertical side of each of the two removed rectangles equals to γ⃗. Then we identify the horizontal sides d_{1} to d_{1} and d_{2} to d_{2}. We get a pair of tori, each provided with a pair of holes.

Finally, we join the hole \overline{w_{1}w′_{1}} on the first torus to the hole \overline{z_{2}z′_{2}} on the second torus by a cylinder (on the picture the cylinder is represented by a narrow stripe with identified opposite sides c_{1} to c_{1}); we also join the hole \overline{w_{2}w′_{2}} on the second torus to the hole \overline{z_{1}z′_{1}} on the first torus by a cylinder (on the picture this second cylinder is represented by a narrow stripe with identified opposite sides c_{2} to c_{2}).

We get a flat surface of genus g = 3 with four simple zeroes. By construction the surface contains two cylinders filled with closed regular geodesics homologous to γ⃗ (these cylinders are represented by narrow stripes with horizontal sides c_{1} and c_{2}). In other words the saddle connection represented by γ has multiplicity 2. Note that the two cylinders would survive under any sufficiently small deformation of the surface, which means that this configuration is generic.

10.3. Admissible Constructions. Consider the following data:

• For 1 ≤ i ≤ p: surfaces S′_{i} ∈ H(α_{i}) with area S′_{i} < 1.

• A vector γ ∈ ℝ^{2}, with |γ| < ϵ.

• For 1 ≤ i ≤ p: a pair of points z_{i}, w_{i} ∈ S′_{i} which are zeroes of orders b′_{i}, b′′_{i} or marked points. The points may coincide. If they are distinct, we are also given a transversal ρ_{i} joining z_{i} and w_{i}. The ρ_{i} may be chosen so that for some fixed M and fixed homology basis of curves β,

(18)  |β · ρ_{i}| ≤ M

where β · ρ_{i} is the algebraic intersection number.

• If w_{i} = z_{i} and is a zero of order a_{i}, then a pair of numbers a′_{i}, a′′_{i} with a_{i} = a′_{i} + a′′_{i}.

• A subset J = {i_{1}, …, i_{q}} ⊂ {1, …, p}, where 0 ≤ q ≤ p. By convention, when J = ∅ we let q = 0.

• For 1 ≤ j ≤ q: a number h_{i_{j}} > 0 such that ∑ h_{i_{j}}|γ| + area S_{i_{j}} < 1.

• For 1 ≤ j ≤ q: a number t_{i_{j}} with 0 ≤ t_{i_{j}} < |γ|. Given the above data we construct a flat surface S with homologous saddles

of multiplicity m with associated data (J, a′_{i}, a′′_{i}, b′_{k}, b′′_{k}). Namely, for each i, if z_{i} = w_{i} we use the figure eight construction and if z_{i} ≠ w_{i} we use the parallelogram construction giving two boundary curves γ′_{i} = \overline{z_{i}z_{i+1}} and γ′′_{i} = \overline{w_{i}w′_{i}}. For each i either glue γ′_{i} directly to γ′′_{i+1} or, if i = i_{j} ∈ {i_{1}, …, i_{q}} = J glue in cylinders C_{i_{j}} of “width” |γ|, “height” h_{i_{j}} and “twist” t_{i_{j}} to connect \overline{w_{i_{j}}w′_{i_{j}}} to \overline{z_{i_{j}+1}z′_{i_{j}+1}} (i + 1 considered mod p).

The resulting surface S also depends on the data h_{i_{j}}, t_{i_{j}} and ρ_{i}. Denote by S′ the union of the pieces S′_{i}, h the vector of h_{i_{j}}, t the vector of t_{i_{j}} and ρ the collection of ρ_{i}. Then we have an admissible construction

(S′, J, a′_{i}, a′′_{i}, b′_{i}, b′′_{i}, ρ, γ, h, t) → S

A relative homology basis for S is given by:

• A relative homology basis of curves β for each component S′_{i} of S′.

• The curve ρ_{i} on each component S′_{i}.

• A “horizontal” curve on each cylinder C or closed curve. The integral of ω along this curve is γ.

• A “vertical” curve on each cylinder. The integral of ω along this curve is

h + ir. The holonomy vectors v_{S}(β) on S of the set of curves β that live on S′_{i} is given

by

(19)  v_{S}(β) = v_{S′}(β) + (β · ρ_{i})γ

Lemma 10.2. Suppose that we

— either use at least one parallelogram construction; or — if we use only figure eight constructions, we use at least one cylinder. Then the resulting surface S is a nondegenerate flat surface.

Given the admissible construction (S′, J, a′_{i}, a′′_{i}, b′_{k}, b′′_{k}, ρ, γ, h, t) → S let us study in more detail how the resulting flat surface S is made up of subsurfaces. The resulting surface S has two groups of zeroes: the ones inherited from the components S′_{i} without any changes, and the newborn ones. Every newborn zero has at least one curve γ_{i} passing through it; the curves γ_{i} do not pass through the zeroes of the first group. Actually, it is easy to see from our construction, that the collection of curves γ_{i_{j}} passing through a zero of S has the form γ_{i}, γ_{i+1}, …, γ_{i+s}, with consecutive indices (where as usual p + 1 is identified with 1, so p is followed by 1). It is easy to give a description of the nature of the corresponding set {i, i + 1, …, i + s}.

Any newborn zero is formed by one of the following constructions:

• Type I. There is a chain of consecutive surfaces S_{j_{r}+1} → S_{j_{r}+2} → ⋯ → S_{j_{r}+s} glued directly. To each surface of the chain we apply the figure eight construction. The first surface of the chain S_{j_{r}+1} is glued to the preceding surface S_{j_{r}} by a cylinder and similarly for the last surface of the chain S_{j_{r}+s} and its successor. In our notation we have j_{r} + s = j_{r+1}, where both j_{r}, j_{r+1} ∈ J, and s ≥ 0.

In this case the newborn zero has order

(20)  ∑_{k=1}^{s}(a_{j_{r}+k} + 2),

where a_{j_{r}+k} = a′_{j_{r}+k} + a′′_{j_{r}+k}. By convention s = 0 stands for the case when multiplicity p is equal to 1, J = {1}, and we have a single surface S_{1} to which we apply figure eight construction and glue in a cylinder.

Figure 10. Newborn zeroes of three different types.

• Type II. Now we have the following chain of surfaces S_{j_{r}+1}, S_{j_{r}+2}, …, S_{j_{r}+s+1} glued directly. We apply the figure eight construction to all surfaces except the last, to which we apply the parallelogram construction. The first surface S_{j_{r}+1} of the chain is glued to S_{j_{r}} by a cylinder. The value s = 0 is allowed: in this case we have a single surface S_{j_{r}+1} to which we apply a parallelogram construction, and then attach a cylinder to the zero z_{j_{r}+1} of order b′_{j_{r}+1} + 1. In this case the newborn zero has order

(21)  ∑_{k=1}^{s}(a_{j_{r}+k} + 2) + (b′′_{j_{r}+s+1} + 1)

We may have the symmetric picture, when the surface S_{j} to which we apply the parallelogram construction is at the beginning of the chain, then we glue directly the surfaces S_{j+1}, …, S_{j+s} to which we apply figure eight construction, and to the last surface S_{j+s} (where j + s ∈ J equals some j_{r}) we glue a cylinder. We have a symmetric formula for the order of the zero:

(b′_{j} + 1) + ∑_{k=1}^{s}(a_{j+k} + 2)

• Type III. Finally, we may have surfaces S_{k}, …, S_{k+s+1} to which we apply the parallelogram construction boarding both ends of the chain; we apply the figure eight construction to all interior surfaces S_{k+1} … S_{k+s}; we glue all surfaces directly. The value s = 0 is admissible: by convention it means that we have only two surfaces S_{k} and S_{k} + 1 in the chain to which we apply parallelogram constructions and then glue directly γ′′_{k} to γ′_{k+1}. In this case we get the newborn zero of the following order

(22)  (b′_{k} + 1) + ∑_{i=1}^{s}(a_{k+i} + 2) + (b′′_{k+s+1} + 1)

We now adopt the following notation to describe how the surfaces are put together.

Notation The symbol → corresponds to direct gluing of the surfaces; the symbol ⇒ corresponds to a cylinder joining two consecutive surfaces. If a number is written as a sum of two numbers a′_{i} + a′′_{i} with a bar over the sum, we apply the figure eight construction to a zero of order a_{i} = a′_{i} +a′′_{i} breaking it up into zeroes of a′_{i} and a′′_{i}; if there are two multiplicity numbers, b̄′_{i}, b̄′′_{i}, we apply the parallelogram construction. By convention b′_{i} is the left one, b′′_{i} is the right one. Say, in (4, ¯3, 2, ¯1) we have b′_{1} = 3, b′′_{1} = 1, and in (\overline{0 + 0}) we have a_{2} = a′_{2} = a′′_{2} = 0.

Example 10.3. Consider the following collection of flat surfaces organized in a cyclic order

→(4,¯3,2,¯1)→(\overline{0+0})→(\overline{4+2},2)→(¯9,8,¯7)→(¯5, ¯3)⇒(\overline{2+2},2)→(5,\overline{1+2})⇒(\overline{1+0},1)→(2,\overline{0+0})→
→ S_{1} → S_{2} → S_{3} → S_{4} → S_{5} ⇒ S_{6} → S_{7} ⇒ S_{8} → S_{9} →

In the example above we glue S_{5} to S_{6} with a cylinder, and S_{7} to S_{8} with a cylinder; all the other gluings are direct. Thus in this example we have p = 9, and J = {5, 7}. The chain S_{1} → S_{2} → S_{3} → S_{4} gives a newborn zero of type III. It has order (1 + 1) + ((0 + 2) + (6 + 2)) + (9 + 1) = 22; there are 3 separatrix loops γ_{1}, γ_{2}, γ_{3} passing through this newborn zero. The loops γ_{1} and γ_{2} bound the surface S_{2} and bound angles 2π with each other. The loops γ_{2}, γ_{3} bound S_{3} with angles 10π and 6π. The next chain S_{4} → S_{5} gives a newborn zero of type III; it has order (7+1)+(5+1) = 14; there is only one separatrix loop γ_{4} passing through this zero. The next chain is S_{5} ⇒. It is of type II; it gives a newborn zero of order 3 + 1 = 4 with a separatrix loop γ_{5} passing through it. The next chain ⇒ S_{6} → S_{7} ⇒ gives a newborn zero of order (4 + 2) + (3 + 2) = 11 of type I, and three separatrix loops γ_{6}, γ_{7}, γ_{8} passing through it. The surface S_{6} is bounded by γ_{6} and γ_{7} which make angles 8π and 4π at the zero of order 11, while S_{7} is bounded by γ_{7} and γ_{8} which make angles 6π and 4π at the zero. Finally, the last chain ⇒ S_{8} → S_{9} → S_{1} produces a newborn zero of type II. Its order is (1 + 2) + (0 + 2) + (3 + 1) = 9; there are three separatrix loops γ_{9}, γ_{10}, γ_{11} passing through it. The loops γ_{9} and γ_{10} bound the surface S_{8} making angles 4π and 2π with each other while γ_{10} and γ_{11} bound S_{9} making angles 2π and 2π with each other.

The resulting surface inherits the zeroes of orders 4, 2, 2, 8, 2, 5, 1, 2 which are coming unchanged from the components S_{i} (the integers which are not barred.) It also has the newborn zeroes of orders 22, 14, 4, 11, 9. Thus the resulting surface S belongs to the stratum H(22, 14, 11, 9, 8, 5, 4, 4, 2, 2, 2, 2, 1).

10.4. Stratum Interchange and γ → −γ Symmetry. In this section we discuss the possible symmetries of the admissible construction

(S, a_{i}‘, a′′_{i}, b′_{k}, b′′_{k}, ρ, γ, h, t) → S

In many aspects these symmetries are analogous to the symmetries discussed in section 7.4. We again consider two settings: the first one with more restrictions, the second one with fewer restrictions.

Problem 1. We assume that all zeroes z_{1}, …, z_{l} of the surface S are numbered. We fix the zeroes z_{1}, …, z_{r} ∈ S, z_{i} ≠ z_{j}, for i ≠ j, of orders m_{1}, …, m_{r} correspondingly. We assume that the flat surface S possesses exactly s loops of saddle connections γ_{1}, …, γ_{s} homologous to some fixed loop γ. We assume that every loop γ_{i} starts and ends at one of the zeroes z_{1}, …, z_{r}.

Cutting along the loops γ_{i}, i = 1, …, r we perform the surgery as described above decomposing S into the collection S′_{i}. The collection is organized in a cyclic order as described above.

We fix the types α′_{i} of the surfaces S′_{i}, the distinguished zeroes (marked points) w_{i}, z_{i} ∈ S′_{i}, and the numbers (a′_{i}, a′′_{i}), (b′_{k}, b′′_{k}) where both types of pairs are ordered. We fix, whether S′_{j} is joined to S′_{j} by a cylinder, or directly.

We start with the case when we fix also the following data. We assume that z_{1}, …, z_{r} are the newborn zeroes numbered with respect to this cyclic order. We assume that the zeroes z_{r+1}, …, z_{l_{1}} lie in S_{1}, the zeroes z_{l_{1}+1}, …, z_{l_{1}+l_{2}} lie in S_{2}, ..., the zeroes z_{l_{1}+···+l_{p−1}+1}, …, z_{l_{1}+···+l_{p}} lie in S_{p}. Here l = l_{1} + ⋯ + l_{p} = card(α).

Problem 2. Now consider the problem with fewer constraints. We count the number of occurrences of configurations of loops of saddle connections described in the previous problem with the only difference that now we assume all the zeroes to be “anonymous”. In other words, we keep all geometric information on configuration (number r of zeroes involved; number s of loops of homologous saddle connections; types α′_{i} of the surfaces S′_{i}, and the cyclic order of their appearances; numbers (a′_{i}, a′′_{i}), (b′_{j}, b′′_{j}), where both types of pairs are ordered; the way (by means of a cylinder or directly) S′_{i} is joined to S′_{i+1}).

However, if we have two zeroes P_{i}, P_{j} of the same degree on the surface S, and, neither of them is “newborn”, in the setting of Problem 2 we do not distinguish cases, when P_{i} lies in S′_{q} and P_{j} gets to S′_{t}, and the opposite case.

Remark 10.4. Problem 2 may be considered as generalization of Problem 1 to the case when the zeroes of S and of S′_{i} are not numbered.

Remark 10.5. In both settings the H(α′_{i}) may be disconnected. We could also specify the connected component of the H(α′_{i}) when this occurs. We prefer the setting where this data is not specified.

Consider the natural action of the cyclic group of order p on the elements of the assignment organized in a cyclic order. If it has a nontrivial stabilizer we denote it by the same symbol Γ as before. We get a symmetry of order |Γ| which we again call the stratum interchange. Note that this symmetry might be different in the settings of Problem 1 and Problem 2.

Example 10.6. The assignment

(¯1, ¯1, 2) → (¯1, ¯1) ⇒ (¯1, ¯1, 2) → (¯1, ¯1) ⇒

does not possess the stratum interchange symmetry in the setting of Problem 1 since the unchanged zeroes of order 2 are named, and so they identify the components of the type (2, 1, 1). However, in the setting of Problem 2 this configuration has the stratum interchange symmetry of order 2.

The assignment

(¯1, ¯1) → (¯1, ¯1) ⇒ (¯1, ¯1) → (¯1, ¯1) ⇒

possesses the stratum interchange symmetry of order 2 in both settings.

Recall that for saddle connections joining distinct zeroes we also might also have the γ → −γ symmetry. It could appear only when m_{1} = m_{2}, for in this case we have no invariant way of choosing the orientation of the saddle connection. When we have only saddle connections joining a zero to itself, there is no geometric way of choosing the orientation of saddle connection. The two different choices of the orientation give two decompositions of S. Thus we get the action of the group ℤ/2ℤ on the collection of assignments. We call this action the γ → −γ symmetry. This action is easily described in combinatorial terms: one has to change the cyclic order in the assignment to the opposite one, as well as the order in each pair (a′_{i}, a′′_{i}) and (b′_{j}, b′′_{j}). Some assignments may stay invariant under this operation. In this case we say that they have γ → −γ symmetry. This property also depends whether we consider the assignments in the setting of Problem 1 or of Problem 2.

Example 10.7. None of the assignments in Example 10.6 have the γ → −γ symmetry.

An assignment of multiplicity 1 has γ → −γ symmetry if and only if a′ = a′′ when we apply the figure eight construction and if and only if b′ = b′′ when we the apply parallelogram construction.

The assignment

(¯2, ¯2, 2) ⇒ (¯2, ¯2, 2) ⇒

has the γ → −γ symmetry in both settings, while the assignment

(¯2, ¯2, 2) ⇒ (¯2, ¯2, 2) ⇒ (¯2, ¯2, 2) ⇒

has the γ → −γ symmetry only in the setting of Problem 2.

Remark 10.8. In describing the construction (S′, J, a′_{i}, a′′_{i}, b′_{i}, b′′_{i}, ρ, γ, h, t) → S one can consider all possible assignments, and then take into account corresponding symmetries, or one can deal with the classes, eliminating the symmetry whenever it is possible.

We have chosen the second way. For example, the assignment (¯1, ¯0, 1) → (see section 11.2) is symmetric to the assignment (¯0, ¯1, 1) → by the γ → −γ symmetry.

Our main objective is to show now that the degeneration of a surface to a point on the principal boundary and the admissible construction of a surface from a point on the principal boundary are essentially inverse operations. The complicating issue is that both depend on choices of transversals and so they need not be exact inverses. However going from one transversal to another amounts to a change of basis and so can be described.

In particular the admissible construction of S depends on a choice of transversals ρ_{i} for each flat surface S′_{i}. If we fix a pair of zeroes of a given S′_{i} and a transversal joining them, then there may be a large set of other flat structures in H(α′_{i}) for which there is no transversal in the same homotopy class. Thus on different parts of H(α′_{i}) we may be forced to choose different transversals. Thus we need to study how the construction depends on the choice of transversal.

Now fix a small κ. Let F′ denote a compact simply connected subset of H_{1}(α′) such that for S′ ∈ F′ every saddle connection has length at least κ. Its complement has volume O(κ^{2}). Let H_{1}^{ϵ,th}(α, C) denote the subset of H_{1}(α, C) consisting of those flat surfaces such that the saddle connections in the configuration have length at most ϵ and every other saddle has length at least κ.

Choose a relative homology basis for all S′ ∈ F′. The coordinates provided by the holonomy along the homology basis β, allows one to define a metric d on F′. Let M′ = M + M′′ where M is given by (18) and M′′ is given by Lemma 9.3. For ϵ < κ/2 find a finite collection of balls B(x_{j}, r_{j}) centered at x_{j} of radius r_{j} such that

• the smaller balls B(x_{j}, r_{j} − M′ϵ) cover F′

• for each B(x_{j}, r_{j}), for each component S′_{i} of S′ for which z_{i} ≠ w_{i} there is a

fixed class of transversals ρ_{i,j} joining z_{i} to w_{i}.

We may use the cover by balls B(x_{j}, r_{j} − M′ϵ) to define a partition of F′ by a finite collection of sets U_{j} ⊂ B(x_{j}, r_{j} − M′ϵ). Now we can give a well defined assignment

σ : F′ → H_{1}^{ϵ,th}(α, C)

Namely each S′ belongs to a unique S′ ∈ U_{j}. Then use the transversal ρ_{i,j} to assign a collection of surfaces S = σ(S′).

As in the previous section, we have the possibility of a construction for −γ coinciding with a construction with γ and also the possibility of stratum interchange. However even without considering these symmetries, it is possible because of the change of transversals from one set U_{j} to another, that the construction is not 1−1. We prove

Proposition 10.9. There is E′ ⊂ F′ such that • Vol(F′ − E′) = O(ϵ^{2}),

• On a subset of E′ of full volume, σ is 1 − 1 except for the γ → −γ and stratum interchange symmetries.

• Vol(H_{1}^{ϵ,th}(α, C) − σ(E′)) = O(ϵ^{4})

Proof. Choose E′ such that • every point in E′ lies in some ball B(x_{j}, r_{j} − M′ϵ)

• every point in E′′ is at least M′ϵ from the boundary of any ball B(x_{k}, r_{k}). The second condition means that if S′ ∈ E′ ∩ U_{j} then the M′ϵ neighborhood of S′ is contained in U_{j}. It also implies

Vol(F′ − E′) = O(ϵ^{2})

We prove σ is 1 − 1 on almost all of E′ except for possible stratum interchange and γ → −γ symmetries. If S′_{1} ≠ S′_{2} belong to the same U_{j} then the collection of surfaces σ(S′_{1}) and σ(S′_{2}) are constructed with the same transversals and yet differ on holonomy for a set of basis curves. The surfaces can be isomorphic only if the isomorphism does not preserve the basis. If the set of configuration curves are preserved then we have the stratum interchange or γ → −γ symmetry. If they are not preserved, then S′ contains a curve of length at most ϵ and this possibility has been ruled out by the definition of F′. The remaining possibility is if for a fixed S′ two or more of the remaining surfaces in σ(S′) are isomorphic by an isomorphism that preserves the set of curves in the configuration. But such an isomorphism must arise from an automorphism of S′ and such S′ have volume 0.

If S′_{1}, S′_{2} ∈ E′ belong to different U_{j}, then their distance apart in the metric d is at least M′ϵ, and then σ(S′_{1}) and σ(S′_{2}) again differ on the homology basis and we have the same conclusion as before.

We now show the third statement. Define E′′ ⊂ E′ as in the definition of E′ except that we require points to lie in balls B(x_{j}, r_{j} − 2M′ϵ) and be at least 2M′ϵ from the boundary of any ball. Given the transversals τ_{i} from Lemma 9.3, let E be the subset of H_{1}^{ϵ,th}(α, C) consisting of those S whose degeneration yields surfaces S′ in E′′. We have

Vol(H_{1}^{ϵ,th}(α, C) − E) = O(ϵ^{4})

For some j, the resulting surface S′ ∈ U_{j} ∩ B(x_{j}, r_{j} − 2M′ϵ). However since it may not be the case that τ_{i} = ρ_{i,j}, it may not be true that σ(S′) = S. The third statement in the Proposition follows from the claim that there is some S′′ ∈ E′ such that S = σ(S′′). To prove the claim, notice that since B(x_{j}, r_{j}) is a ball and since |β · τ_{i}| ≤ M′′ and |β · ρ_{i,j}| ≤ M, the point in ℝ^{n′} defined by the collection of vectors

v_{S′}(β) + (∑_{i}(β · τ_{i}) − (β · ρ_{i,j}))γ

as β runs over the homology basis, corresponds to the holonomy for a unique S′′ ∈ B(x_{j}, r_{j} − M′ϵ). By definition of E′′, we have S′′ ∈ E′. But now σ(S′′) has the same holonomy as S along the basis and so is isomorphic to it. This proves the claim. □

from S′′′ to S′′ changes distances at most

10.5. Combinatorial Factor, Connected Strata. We need to compute the combinatorial factor M, which computes for fixed data (S′, J, a′_{i}, a′′_{i}, b′_{k}, b′′_{k}, γ, h, t) the number of S = σ(S′) that can be built. We start with the setting of Problem 1 in section 10.4. We first suppose that the stratum H(α) is connected.

Consider a surface S′_{i} such that z_{i} = w_{i} is a zero of order a_{i} = a′_{i} + a′′_{i}. We apply the figure eight construction by breaking z_{i} = w_{i} into zeroes of orders a′_{i} and a′′_{i}. There are a_{i} + 1 ways of doing this, see section 10.1. The corresponding boundary components γ′_{i}, γ′′_{i} are octagon. The component γ′_{i} is attached to S_{i+1} (directly or by means of a cylinder depending on whether or not i ∈ J); the component γ′′_{i} is attached to S_{i−1} (directly or by means of a cylinder depending on whether or not i − 1 ∈ J).

If z_{k} ≠ w_{k} we perform the parallelogram construction. The orders of the zeroes are b′′_{k} and b′_{k} correspondingly, so there are b′′_{k} + 1 ways of choosing the vector γ at z_{k} and b′_{k} + 1 ways of choosing the vector γ at w_{i} so this gives (b′_{k} + 1)(b′′_{k} + 1) choices.

We may have the stratum interchange symmetry or (and) γ → −γ symmetry, see section 10.4. Thus the resulting combinatorial factor in the setting of Problem 1 in section 10.4 equals

c = \frac{1}{|Γ_{−}|} · \frac{1}{|Γ|} · ∏_{\begin{matrix}1≤i≤p \\ z_{i}=w_{i} \end{matrix}}(a_{i} + 1) · ∏_{\begin{matrix}1≤k≤p \\ z_{k}≠w_{k} \end{matrix}}(b′_{k} + 1)(b′′_{k} + 1)

We now compute the combinatorial factor in the setting of Problem 2 in section 10.4. For each integer m, let o(m) denote its multiplicity. For each i, i = 1, …, p, let o_{i}(m) denote the multiplicity of m in α′_{i}.

Consider an integer m different from the order of any newborn zero on S (see formulae 20, 21, 22), and different from any a_{i} or b′_{k}, b′′_{k}. Since all the zeroes z_{1}, z_{2}, …, z_{l} of S are numbered, the number of ways to arrange o(m) zeroes of order m into groups of o_{1}(m), …, o_{p}(m) zeroes equals

\frac{o(m)!}{∏_{i=1}^{p} o_{i}(m)!}

where we let 0! = 1 by convention.

Let now s ≥ 1 be the number of the newborn zeroes in S of order m, and suppose that m is different from any a_{i}, b′_{k}, b′′_{k}. Now we have to chose s of o(m) numbered zeroes to be the newborn ones, and then we have to arrange the remaining o(m)−s ordered zeroes into groups of o_{1}(m), …, o_{p}(m) zeroes. For such m we get the factor

o(m)(o(m) − 1) · · · · · (o(m) − s + 1) · \frac{(o(m) − s)!}{∏_{i=1}^{p} o_{i}(m)!} = \frac{o(m)!}{∏_{i=1}^{p} o_{i}(m)!}

which coincides with the previous one. Hence, we do not need to distinguish this case from the previous one.

If m is different from any b′_{k}, b′′_{k}, but is equal to some a_{i}, then there are only o_{i}(m) − 1 zeroes of order m on S′_{i} inherited from S. Thus the corresponding factor in the denominator in the formula above equals (o_{i}(m)−1)!. Multiplying numerator and denominator by o_{i}(m) we get the following factor for such m:

\frac{o(m)!}{∏_{i=1}^{p} o_{i}(m)!} · ∏_{\begin{matrix}1≤i≤p \\ a_{i}=m \\ z_{i}=w_{i} \end{matrix}} o_{i}(m)

We have to make similar correction for those m which coincide with some b′_{k} or b′′_{k}. In the case when m equals one of the b′_{k} ≠ b′′_{k} the correction is completely analogous to the previous one. In the case when for some k we have m = b′_{k} = b′′_{k}, we get only o(m) − 2 zeroes of order m on S′_{k}, inherited from S. Thus the corresponding factor in the denominator in the formula above equals (o_{k}(m) − 2)!. Multiplying numerator and denominator by o_{k}(m)(o_{k}(m) − 1) and collecting all the correction terms we finally get the following combinatorial factor for the setting of Problem 2 in section 10.4.

M = \frac{1}{|Γ_{−}|} · \frac{1}{|Γ|} · ∏_{m∈α} (\frac{o(m)!}{∏_{i=1}^{p} o_{i}(m)!}) ·

· ∏_{\begin{matrix}1≤i≤p \\ z_{i}=w_{i} \\ a_{i}≠0 \end{matrix}} o_{i}(a_{i}) · ∏_{\begin{matrix}1≤k≤p \\ z_{k}≠w_{k} \\ b′_{k}≠b′′_{k} \\ b′_{k}≠0 \end{matrix}} o_{k}(b′_{k}) · ∏_{\begin{matrix}1≤k≤p \\ z_{k}≠w_{k} \\ b′_{k}≠b′′_{k} \\ b′′_{k}≠0 \end{matrix}} o_{k}(b′′_{k}) · ∏_{\begin{matrix}1≤k≤p \\ z_{k}≠w_{k} \\ b′_{k}=b′′_{k}≠0 \end{matrix}} o_{k}(b′_{k})(o_{k}(b′_{k}) − 1)·

· ∏_{\begin{matrix}1≤i≤p \\ z_{i}=w_{i} \end{matrix}}(a_{i} + 1) · ∏_{\begin{matrix}1≤k≤p \\ z_{k}≠w_{k} \end{matrix}}(b′_{k} + 1)(b′′_{k} + 1)

Note that in general the symmetry groups Γ and Γ_{−} corresponding to possible stratum interchange symmetry and to γ → −γ symmetry are different in Problems 1 and 2, see section 10.4.

There will also be additional factors coming from the spin structures and the hyperelliptic strata. We shall discuss these separately.

11. Computing the Constants for Connected Strata

Recall that q is the number of cylinders we shall attach. Let n_{i} = dim_{ℝ} H(α_{i}). We also define d_{i} to equal n_{i} except in the following situations. If α_{i} has a single marked point which is not a zero and S_{i} is not the torus, we have d_{i} = n_{i} + 2. If there are a pair of marked point, and the surface is not the torus d_{i} = n_{i} + 4. If the surface is a torus then d_{i} = n_{i} + 2. Then,

dim_{ℝ} H(α) = 2q + 2 + ∑_{i=1}^{p} d_{i},

where α is the stratum such that S ∈ H(α).

11.1. Computation of the constants. Recall that dν(S) is the measure in H(α) and dν(S′_{i}) is the measure in H(α_{i}). We first assume that H(α) is connected. If we choose a homology basis β_{i} for the S′_{i}, and let hol_{S}(β_{i}) be the holonomy of β_{i} on the surface S, then the measure dν restricted to the image σ(E′) under the admissible construction is the product measure of Mdγ dt_{i}dh_{i} and ∏ d(hol_{S}(β_{i})). By Equation (19) the latter coincides with dν′ up to an error that is O(ϵ^{2}). Since |γ| ≤ ϵ, up to an error O(ϵ^{4}) we can write

dν(S) = Mdγ ∏_{i=1}^{p} dν′(S′_{i}) ∏_{i=1}^{q} dh_{i}dt_{i}

except for the following situations. If z_{i} is a marked point and w_{i} is a zero then we have the additional factor dz(S_{i}) in the term dν_{i}. If z_{i} is a zero and w_{i} is a marked point we have dw(S_{i}). If w_{i} = z_{i} is a marked point, we have dz(S_{i}) and if z_{i} ≠ w_{i} are both marked points we have the factor dz(S_{i}) dw(S_{i}).

Note that we could also consider strata with one or two marked points. The stratum would be a fiber space over the stratum H_{1}(α). The fiber over a point would be the flat surface if there is one marked point or product of flat surfaces if there are two, the flat surface representing that point in H_{1}(α). The measure on the fiber space is then locally product measure of the base with the area (or product of area measures) on the fiber.

Note that (area S) = ∑_{i}(area S_{i}) + h_{i}|γ|, and if S ∈ C(H_{1}^{ϵ,th}(α, C)) then |γ| ≤ ϵ\sqrt{(area S)}. Hence,

|γ|^{2} ≤ ϵ^{2} ∑_{i}(area S_{i}) + h_{i}|γ|,

i.e.

∑_{i} h_{i} ≥ \frac{|γ|}{ϵ^{2}} − \frac{∑_{i}(area S_{i})}{|γ|}.

The expression on the left is positive when |γ| > ϵ\sqrt{(∑_{i} area S_{i})}.

Let

W = ∏_{i} Vol(H_{1}(α_{i}))

For s > 0, let D(s) = {(r_{1}, …, r_{p}) : ∑_{i} r_{i}^{2} < s}. Then by Proposition 10.9, and the fact that Vol(∏_{i} H_{1}(α_{i})−F′) = O(κ^{2}), up to an error term that is O(ϵ^{2}(ϵ^{2}+κ^{2})) we have

ν(C(H_{1}^{ϵ,th}(α))) = W M·

= ( ∫_{D(1)}(∏_{i=1}^{p} r_{i}^{d_{i}−1} dr_{i}) ∫_{γ∈B(ϵ\sqrt{∑_{i} r_{i}^{2}})} ∫_{0≤∑_{i} h_{i}≤^{\frac{1−∑_{i} r^{2}_{i}}{|γ|}}} ∏_{i=1}^{q}(∫_{0}^{|γ|} dt_{i} dh_{i} )dγ+

+ ∫_{D(1)} ∏_{i=1}^{p} r_{i}^{d_{i}−1} dr_{i} ∫_{γ∈B(ϵ)−B(ϵ\sqrt{∑_{i} r_{i}^{2}})} ∫_{\frac{|γ|}{ϵ^{2}} −^{\frac{∑_{i} r^{2}_{i}}{|γ|}} ≤∑_{i} h_{i}≤^{\frac{1−∑_{i} r^{2}_{i}}{|γ|}}} |γ|^{q}dh_{i} dγ) =

\begin{matrix}= W M · ( ∫_{D(1)} ∏_{i=1}^{p} r_{i}^{d_{i}−1} dr_{i} ∫_{γ∈B(ϵ\sqrt{∑_{i} r_{i}^{2}})} \frac{1}{q!}(1 − ∑_{i} r_{i}^{2})^{q} dγ+ \\ + ∫_{D(1)}∏_{i=1}^{p} r_{i}^{d_{i}−1} dr_{i} ∫_{γ∈B(ϵ)−B(ϵ\sqrt{∑_{i} r_{i}^{2}})} \frac{1}{q!}[(1 − ∑_{i} r_{i}^{2})^{q} − (\frac{|γ|^{2}}{ϵ^{2}} − ∑_{i} r_{i}^{2})^{q}]dγ) \end{matrix}

Performing the integral over γ, up to this error term, we obtain

ν(C(H_{1}^{ϵ,th}(α, C))) =

= \frac{πϵ^{2}W M}{q!} ∫_{D(1)} (∏_{i=1}^{p} r_{i}^{d_{i}−1}) (1 − ∑_{i} r_{i}^{2})^{q} (∑_{i} r_{i}^{2}) ∏_{i} dr_{i}

+ \frac{πϵ^{2}W M}{q!} ∫_{D(1)} (∏_{i=1}^{p} r_{i}^{d_{i}−1}) (1 − ∑_{i} r_{i}^{2})^{q} (1 − ∑_{i} r_{i}^{2}) ∏_{i} dr_{i}

− \frac{πW M}{q!} ∫_{D(1)} (∏_{i=1}^{p} r_{i}^{d_{i}−1}) ∫_{ϵ\sqrt{∑_{i} r_{i}^{2}}}^{ϵ} (\frac{s^{2}}{ϵ^{2}} − ∑_{i} r_{i}^{2})^{q} 2s ds ∏_{i} dr_{i}

= \frac{πϵ^{2}W M}{q!} ∫_{D(1)} (∏_{i=1}^{p} r_{i}^{d_{i}−1}) (1 − ∑_{i} r_{i}^{2})^{q} ∏_{i} dr_{i}

− \frac{πϵ^{2}W M}{(q + 1)!} ∫_{D(1)} (∏_{i=1}^{p} r_{i}^{d_{i}−1}) (1 − ∑_{i} r_{i}^{2})^{q+1} ∏_{i} dr_{i}

where we have used the change of variable u = (s^{2}/ϵ^{2} − ∑ r_{i}^{2}) in the last line. We now make the change of variable x_{i} = r_{i}^{2}. Let b_{i} = d_{i}/2 − 1, so that r_{i}^{d_{i}−1} dr_{i} = (1/2)x_{i}^{b_{i}} dx_{i}. Then,

ν(C(H_{1}^{ϵ,th}(α, C))) = \frac{πϵ^{2}W M}{2^{p}} ∫_{∑_{i} x_{i}≤1} x_{1}^{b_{1}} … x_{p}^{b_{p}}[\frac{1}{q!} (1 − ∑_{i} x_{i})^{q}

− \frac{1}{(q + 1)!} (1 − ∑_{i} x_{i})^{q+1} ] dx_{1} … dx_{p}

Evaluating the integral from the inside out via repeated application of the identity

∫_{0}^{u} x^{a}(u − x)^{b} dx = \frac{a! b!}{(a + b + 1)!}u^{a+b+1}

yields

ν(C(H_{1}^{ϵ,th}(α, C))) =

= \frac{πϵ^{2}W M}{2^{q}} [\frac{b_{1}! … b_{q}!}{(b_{1} + ⋯ + b_{p} + p + q)!} − \frac{b_{1}! … b_{p}!}{(b_{1} + ⋯ + b_{p} + p + q + 1)!}] =

= \frac{πϵ^{2}W M}{2^{p}} \frac{b_{1}! … b_{p}!}{(b_{1} + ⋯ + b_{p} + p + q − 1)!} · \frac{1}{b_{1} + ⋯ + b_{p} + p + q + 1}

Since b_{1} + ⋯ + b_{p} + p + q + 1 = n/2, up to an error term that is O(ϵ^{2}(ϵ^{2} + κ^{2})), we have

ν(C(H_{1}^{ϵ,th}(α, C))) = \frac{πϵ^{2}W M}{2^{p−1}} \frac{b_{1}! … b_{p}!}{(\frac{n}{2} − 2)!}

Now first taking the limit as ϵ → 0 for fixed κ, and then taking the limit as κ → 0, we get

c = lim_{κ→0} lim_{ϵ→0} \frac{ν(C(H_{1}^{ϵ}(α, C)))}{πϵ^{2}ν_{1}(H_{1}(α))} =

= M · \frac{1}{2^{p−1}} · \frac{∏_{i=1}^{p}(\frac{d_{i}}{2} − 1)!}{(\frac{n}{2} − 2)!} · \frac{∏_{i=1}^{p} Vol(H_{1}(α′_{i}))}{Vol(H_{1}(α))}

We finally get the following expression for the constant c in the setting where the zeroes of S are not numbered (see Problem 2 in section 10.4):

Formula 11.1. The list of possible configurations of homologous saddle connections joining zeroes to themselves is the same for almost all surfaces in any connected stratum H(α). The possible configurations are described in section 10.3. The constant in the quadratic asymptotics for the number of saddle connections with associated data (J, a′_{i}, a′′_{i}, b′_{k}, b′′_{k}) is presented by the following formula:

(23) c = \frac{1}{|Γ_{−}|} · \frac{1}{|Γ|} · ∏_{m∈α} (\frac{o(m)!}{∏_{i=1}^{p} o_{i}(m)!}) ·

· ∏_{\begin{matrix}1≤i≤p \\ z_{i}=w_{i} \\ a_{i}∈α \end{matrix}} o_{i}(a_{i}) · ∏_{\begin{matrix}1≤k≤p \\ z_{k}≠w_{k} \\ b′_{k}≠b′′_{k} \\ b′_{k}∈α \end{matrix}} o_{k}(b′_{k}) · ∏_{\begin{matrix}1≤k≤p \\ z_{k}≠w_{k} \\ b′_{k}≠b′′_{k} \\ b′′_{k}∈α \end{matrix}} o_{k}(b′′_{k}) · ∏_{\begin{matrix}1≤k≤p \\ z_{k}≠w_{k} \\ b′_{k}=b′′_{k}∈α \end{matrix}} o_{k}(b′_{k})(o_{k}(b′_{k}) − 1)·

· ∏_{\begin{matrix}1≤i≤p \\ z_{i}=w_{i} \end{matrix}}(a_{i} + 1) · ∏_{\begin{matrix}1≤k≤p \\ z_{k}≠w_{k} \end{matrix}}(b′_{k} + 1)(b′′_{k} + 1)·

· \frac{1}{2^{p−1}} · \frac{∏_{i=1}^{p}(\frac{d_{i}}{2} − 1)!}{(\frac{d}{2} − 2)!} · \frac{∏_{i=1}^{p} Vol(H_{1}(α′_{i}))}{Vol(H_{1}(α))}

11.2. Examples: Constants for Strata in Genus 3. Stratum H(3, 1)

→ (\overline{1}, \overline{0}, 1) → The first possibility is multiplicity 1 with no cylinder. After degeneration we get a genus 2 surface with a distinguished simple zero, a marked point and another simple zero. We have d_{1} = 12. We employ the parallelogram construction to the distinguished simple zero and marked point producing two circles which we glue directly. Since 1 = b′_{1} ≠ b′′_{1} = 0 we do not have the γ → −γ symmetry. Thus M = (b′_{1} + 1)(b′′_{1} + 1) = 2. We have

c = 2 · \frac{(5 + 1 − 1)!}{(7 − 2)!} · \frac{Vol(H_{1}(1, 1))}{Vol(H_{1}(3, 1))} = \frac{105}{16} · \frac{1}{ζ(2)} ≈ 3.990

⇒ (\overline{1 + 0}, 1) ⇒ The next possibility is multiplicity 1 with a cylinder where the other side also returns to the zero of order 3. After degeneration we get a surface of genus 2 with a distinguished simple zero on which we perform the figure eight construction, gluing in a cylinder. Since 1 = a′_{1} ≠ a′′_{1} = 0 we do not have the γ → −γ symmetry. Thus M = (a_{1} + 1) = 2. Now d_{1} = 10 so

c = 2 · \frac{(5 − 1)!}{(7 − 2)!} · \frac{Vol(H_{1}(1, 1))}{Vol(H_{1}(3, 1))} = \frac{21}{16} · \frac{1}{ζ(2)} ≈ 0.7979

⇒ (\overline{2}, \overline{0}) ⇒ If the other side returns to the simple zero, then after degeneration, we get a genus 2 surface with a double zero and a marked point on which we perform the parallelogram construction. Again d_{1} = 10. Since 2 = b′_{1} ≠ b′′_{1} = 0 we do not have the γ → −γ symmetry. Thus M = (b′_{1} + 1)(b′′_{1} + 1) = 3. We have

c = 3 · \frac{(4 + 1 − 1)!}{(7 − 2)!} · \frac{Vol(H_{1}(2))}{Vol(H_{1}(3, 1))} = \frac{567}{256} · \frac{1}{ζ(2)} ≈ 1.346

⇒ (\overline{0 + 0}) → (\overline{0}, \overline{0}) ⇒ Now we consider multiplicity 2 in this stratum. Then we have a cylinder whose boundary components are curves returning to the zero of order 3 and zero of order one. The homologous curve returns to the zero of order 3. The degenerating surfaces are tori, one with a single marked point, the other with 2 marked points. On the torus with one point, we perform the figure eight construction and on the other the parallelogram construction. We glue in one cylinder and one pair of circles is glued directly.

There is no γ → −γ symmetry in this case.

In this case we get

c = 1 · \frac{1}{2^{2−1}} · \frac{(3 − 1)! · (2 − 1)!}{(7 − 2)!} · \frac{Vol(H_{1}(∅)))^{2}}{Vol(H_{1}(3, 1))} = \frac{105}{256} · \frac{1}{ζ(2)} ≈ 0.2493

Stratum H(2, 1, 1)

→ (1, 1, \overline{0}, \overline{0}) → The multiplicity 1 case with no cylinder degenerates to a genus 2 surface with 2 marked points. We apply the parallelogram construction to the marked points, gluing the circles directly to each other. There is the γ → −γ symmetry.

We get

c = \frac{1}{2} · \frac{(5 + 2 − 1)!}{(8 − 2)!} · \frac{Vol(H_{1}(1, 1))}{Vol(H_{1}(2, 1, 1))} = \frac{7}{3} · \frac{1}{ζ(2)} ≈ 1.418

⇒ (1, 1, \overline{0 + 0}) ⇒ For the case of a cylinder with multiplicity 1 where the other side returns to the double zero the degenerating surface has genus 2 with a single marked point. We apply figure The eight construction at the marked point. We have the γ → −γ symmetry.

c = \frac{1}{2} · \frac{(5 + 1 − 1)!}{(8 − 2)!} · \frac{Vol(H_{1}(1, 1))}{Vol(H_{1}(2, 1, 1))} = \frac{7}{18} · \frac{1}{ζ(2)} ≈ 0.2364

⇒ (1, \overline{1}, \overline{0}) ⇒ If the other side of the cylinder returns to a simple zero, then after degeneration we get a surface of genus 2 with a marked point and a simple zero (as well as another simple zero on the surface). We apply the parallelogram construction at the marked point and a zero gluing in a cylinder. Now we have b′_{1} = 0, b′′_{1} = 1. Since b′_{1} ≠ b′′_{2} we do not have a γ → −γ symmetry.

c = 4 · \frac{(5 + 1 − 1)!}{(8 − 2)!} · \frac{Vol(H_{1}(1, 1))}{Vol(H_{1}(2, 1, 1))} = \frac{28}{9} · \frac{1}{ζ(2)} ≈ 1.891

⇒ (2, \overline{0}, \overline{0}) ⇒ If the cylinder joins two simple zeroes, then after degeneration we get a surface of genus 2 with a zero of order 2 and two marked points. We apply the parallelogram construction at the marked points. We have b′_{1} = b′′_{1} = 0. Here we have γ → −γ symmetry.

c = 1 · \frac{(4 + 2 − 1)!}{(8 − 2)!} · \frac{Vol(H_{1}(2))}{Vol(H_{1}(2, 1, 1))} = \frac{7}{8} · \frac{1}{ζ(2)} ≈ 0.5319

⇒ (\overline{0}, \overline{0}) → (\overline{0}, \overline{0}) ⇒ There are two multiplicity p = 2 cases. The first is if there is a cylinder with boundary curves returning to the simple zeroes and a homologous curve returning to the double zero. We get 2 tori each with 2 marked points so p = 2, d_{1} = d_{2} = 6. We perform the parallelogram construction on each and then glue in one cylinder and glue two circles directly. We have the γ → −γ symmetry, and we do not have a stratum interchange.

c = 1 · \frac{1}{2^{2−1}} · \frac{(3 − 1)! · (3 − 1)!}{(8 − 2)!} · \frac{Vol(H_{1}(∅))^{2}}{Vol(H_{1}(2, 1, 1))} = \frac{7}{36} · \frac{1}{ζ(2)} ≈ 0.1182

⇒ (\overline{0}, \overline{0}) ⇒ (\overline{0 + 0}) ⇒ The second case is if there are 2 cylinders. In that case, after degeneration, we get a torus with 2 marked points and a torus with one marked point. We apply the figure eight construction to the torus with one marked point and the parallelogram construction to the other. We then glue in cylinders to each pair of circles. Again we have the γ → −γ symmetry, and we do not have a stratum interchange.

c = 1 · \frac{1}{2^{2−1}} · \frac{(3 − 1)! · (2 − 1)!}{(8 − 2)!} · \frac{Vol(H_{1}(∅))^{2}}{Vol(H_{1}(2, 1, 1))} = \frac{7}{72} · \frac{1}{ζ(2)} ≈ 0.05910

11.3. Principal stratum. Since any figure eight construction produces a “newborn” zero of order at least two, here we cannot have any figure eight constructions. All the surfaces S′_{i}, belong to the principal strata in lower genera g_{i}; there are a pair of marked points z_{i} ≠ w_{i} on each surface, and we always apply the parallelogram construction with b′_{i} = b′′_{i} = 0. All the curves γ_{i} bound cylinders.

11.3.1. Saddle Connections of Multiplicity 1. In multiplicity one there is a single surface S′ ∈ H(α′) of genus g − 1 and S′ has two marked points z ≠ w. We recover S from S′ by applying the parallelogram construction followed by gluing in a cylinder; α′ is the partition of 2(g − 1) − 2 into ones.

We always have the γ → −γ symmetry, so |Γ_{−}| = 2 in all cases. We use the setting of Problem 2 of section 10.4). The combinatorial constant is

M = \frac{1}{2} · \frac{(2g − 2)!}{(2g − 4)!} · (0 + 1)(0 + 1)

Applying (23) we get the following value for the constant c:

c = \frac{(2g − 2)(2g − 3)}{2} · \frac{((4g − 3 − 2) − 1)!}{((4g − 3) − 2)!} · \frac{Vol(H_{1}(α′))}{Vol(H_{1}(α))}

Thus, finally,

c = \frac{(g − 1)(2g − 3)}{4g − 5} · \frac{Vol(H_{1}(\overbrace{1, …, 1}^{2g−4}))}{Vol(H_{1}(\underbrace{1, …, 1}_{2g−2}))}

In the table below we present the values of the constant for saddle connections of multiplicity one joining a zero to itself, for flat surfaces living in the principal strata in small genera.

Table 7. Principal stratum H(1, …, 1); values of the constants in the quadratic asymptotics for the number of closed geodesics of multiplicity one.

g = 2  g = 3  g = 4  g = 5  g = 6  g = 7  g = 8
c · ζ(2) =  \frac{5}{2}  \frac{36}{7}  \frac{3150}{377}  \frac{274456}{23357} \frac{250153470}{16493303} \frac{6531347988}{351964697} \frac{8007196856750}{365342975469}
c ≈  1.51982 3.12648 5.07950 7.14344  9.22041  11.2812  13.3239

11.3.2. Higher Multiplicity in the Principle Stratum. The surface S breaks up into a union of p surfaces S_{i}. of genus g_{i}. The S_{i} have simple zeroes. Let α_{i} denote the partition of 2(g_{i} − 2) into ones. Each S_{i} has two marked points z_{i} and w_{i}.

We reconstruct S by the parallelogram construction and then by gluing in cylinders C_{i} to connect the boundary circles.

There is again always a factor of 1/2 due to the symmetry γ → −γ since all points are marked points. The combinatorial factor is

M = \frac{(2g − 2)!}{∏_{i=1}^{p}(2g_{i} − 2)!}

If all S_{i} lie in the same stratum then we have the further stratum interchange symmetry

S_{1} → S_{2} → … S_{p} →

and we need to further divide by p in this case.

Thus applying our formula with M as above, we have

c = \frac{M}{2^{p−1}(4g − 5)! Vol(H_{1}(α))} ∏_{i=1}^{p}(4g_{i} − 2)! Vol(H_{1}(α_{i}))

To complete this section we present some examples of these computations.

Example 11.1. Stratum H(1, 1, 1, 1). In the stratum H(1, 1, 1, 1) in genus 3, the only higher multiplicity is p = 2 so S′_{1}, S′_{2} have genus 1. We have the stratum interchange symmetry factor of |Γ| = 2; as in all these cases we also have the γ → −γ symmetry, |Γ_{−}| = 2. Hence we have the combinatorial factor

M = \frac{1}{2} · \frac{1}{2} · \frac{4!}{0! · 0!} = 6

Thus

c = 6 · \frac{1}{2^{2−1}} · \frac{2! · 2!}{7!} · \frac{(Vol(H_{1}(∅)))^{2}}{Vol(H_{1}(1, 1, 1, 1))} = \frac{3}{14} · \frac{1}{ζ(2)} ≈ 0.1303

Example 11.2. Stratum H(1, 1, 1, 1, 1, 1). Now g = 4, and we have two different nontrivial partitions of g − 1 = 3 arranged in a cyclic order: in multiplicity 2, a surface of genus 2 and a surface of genus 1 and in multiplicity 3 yielding three tori. In the first case we do not have the stratum interchange symmetry, |Γ| = 1. Hence we get the following combinatorial factor

M = \frac{1}{2} · \frac{6!}{0! · 2!}

Thus

c = \frac{6!}{4} · \frac{1}{2^{2−1}} · \frac{2! · 6!}{11!} · \frac{Vol(H_{1}(∅)) Vol(H_{1}(1, 1))}{Vol(H_{1}(1, 1, 1, 1, 1, 1))} = \frac{90}{377} · \frac{1}{ζ(2)} ≈ 0.1451

In the second case we have the stratum interchange symmetry, |Γ| = 3. Hence we get the following combinatorial factor

M = \frac{1}{2} · \frac{1}{3} · \frac{6!}{0! · 0! · 0!} = 5!

Thus

c = 5! · \frac{1}{2^{3−1}} · \frac{2! · 2! · 2!}{11!} · \frac{(Vol(H_{1}(∅)))^{3}}{Vol(H_{1}(1, 1, 1, 1, 1, 1))} = \frac{5}{754} · \frac{1}{ζ(2)} ≈ 0.004031

Example 11.3. Stratum H(1, 1, 1, 1, 1, 1, 1, 1). Proceeding to genus 5 we get the following nontrivial partitions of g−1 = 4 arranged in a cyclic order: in multiplicity 2 giving a surface of genus 1 and one of genus 3; in multiplicity 2 yielding two surfaces of genus 2, in multiplicity 3 yielding two tori and a genus 2 surface and in multiplicity 4 yielding 4 tori.

In the first case we do not have any stratum interchange symmetry, |Γ| = 1. Hence we get the following combinatorial factor

M = \frac{1}{2} · \frac{8!}{0! · 4!}

Thus

c = \frac{8!}{2 · 4!} · \frac{1}{2^{2−1}} · \frac{2! · 10!}{15!} · \frac{Vol(H_{1}(∅)) Vol(H_{1}(1, 1, 1, 1))}{Vol(H_{1}(1, 1, 1, 1, 1, 1, 1, 1))} = \frac{4200}{23357} · \frac{1}{ζ(2)} ≈ 0.1093

For the other multiplicity 2 case α′_{1} = α′_{2} = (1, 1). We have the stratum interchange symmetry, |Γ| = 2. Hence we get the following combinatorial factor

M = \frac{1}{2} · \frac{1}{2} · \frac{8!}{2! · 2!}

Thus

c = \frac{8!}{16} · \frac{1}{2^{2−1}} · \frac{6! · 6!}{15!} · \frac{(Vol(H_{1}(1, 1)))^{2}}{Vol(H_{1}(1, 1, 1, 1, 1, 1, 1, 1))} = \frac{720}{23357} · \frac{1}{ζ(2)} ≈ 0.01874

For the third case we do not have any stratum interchange symmetry, |Γ| = 1. Hence we get the following combinatorial factor

M = \frac{1}{2} · \frac{8!}{0! · 0! · 2!}

Thus

c = \frac{8!}{4} · \frac{1}{2^{3−1}} · \frac{2! · 2! · 6!}{15!} · \frac{(Vol(H_{1}(∅)))^{2} Vol(H_{1}(1, 1))}{Vol(H_{1}(1, 1, 1, 1, 1, 1, 1, 1))} = \frac{120}{23357} · \frac{1}{ζ(2)} ≈ 0.003123

In the last case we have stratum a interchange symmetry, |Γ| = 4. Hence we get the following combinatorial factor

M = \frac{1}{2} · \frac{1}{4} · \frac{8!}{0! · 0! · 0! · 0!} = 7!

Thus

c = 7!· \frac{1}{2^{4−1}} · \frac{2! · 2! · 2! · 2!}{15!} · \frac{(Vol(H_{1}(∅)))^{4}}{Vol(H_{1}(1, 1, 1, 1, 1, 1, 1, 1))} = \frac{5}{46714} · \frac{1}{ζ(2)} ≈ 0.00006507

12. Strata that are not Connected

12.1. Parity of the Spin Structure of the Compound Surface. Consider a flat surface S constructed in the previous section. Throughout this section we suppose that all zeroes of ω are of even orders, so the resulting flat surface S has a spin structure. In this section we discuss the conditions on components S′_{i} , and on the gluing rules between them which allow us to obtain a surface S with even α. We also compute the parity of the spin structure of S. This will allow us to find constants in the case of components with spin structures.

Lemma 12.1. Let a nondegenerate flat surface S be obtained from a collection S′_{i} by applying the figure eight and parallelogram constructions. The resulting flat surface S has zeroes of even orders if and only if the following conditions are valid:

— All zeroes of those surfaces S′_{i} to which we apply the figure eight construction have even orders;

— If there is at least one surface to which we apply the parallelogram construction, then there is the following additional alternative. Either

— all zeroes of all surfaces are of even order and every newborn zero of S is of type III;

or

— the orders b′_{k}, b′′_{k} of all zeroes z_{k} ≠ w_{k} to which we apply the parallelogram construction are odd while all the other zeroes of any S_{i} are even.

Proof. First note that all zeroes inherited by S from S′_{i} without changes are of even orders. Thus, if we apply to S′_{i} the figure eight construction, than all the zeroes of S′_{i} different from z_{i} must have even orders. Since the total sum of orders equals 2g_{i} + 2, it means that a_{i} — the order of z_{i} — is even as well.

Suppose now that there is a surface S′_{k} to which we apply the parallelogram construction with at least one of b′_{k}, b′′_{k} even. Without loss of generality we may assume that b′_{k} is even. By the same arguments as above all zeroes of S′_{k} different from z_{k} and w_{k} have even orders. Since the total sum 2g_{k} + 2 of all orders is even, the remaining zero has even order b′′_{k} as well.

Consider the newborn zero of S induced from w′_{k}. It is of one of the types II or III. If it were of type II, then by (21) it would have odd order. Thus this newborn zero is of the type III. Let S′_{k+s+1} be the surface, to which we apply parallelogram construction, at the opposite end of the chain from S′_{k}. By (22) b′_{k+s+1} is even. Repeating our arguments we show that all zeroes of all S′_{i} are even, and that all newborn zeroes are of the type III.

The remaining part of the alternative now becomes obvious. □

Let us calculate the parity of the spin structure of the resulting surface S in all possible cases described above.

Lemma 12.2. Suppose that we use only figure eight constructions to glue a nondegenerate surface S. Suppose that all α_{i} are even. The parity of the spin structure of the resulting surface S is defined by the parities of the spin structures of the components, and by the parities of the numbers a′_{i} in the following way:

ϕ(S) = 1 + ∑_{i=1}^{p} ϕ(S′_{i}) + ∑(a′_{i} + 1) (mod 2)

Proof. For each surface S_{i} consider a collection of 2g_{i} smooth simple closed curves on S′_{i} representing a canonical basis of cycles. As usual we may assume that the curves stay outside a neighborhood of the zeroes and the marked points of S′_{i}.

We may construct now the following canonical basis of cycles on the resulting surface S. Let g be the genus of S. First take all the cycles represented by the basic curves on components. They give us g − 1 pairs of cycles. By construction they form incomplete canonical basis of cycles on S.

Since we are using only figure eight constructions we have to use at least one cylinder to obtain a nondegenerate surface S. As a cycle A_{g} we may use the cycle γ represented by the waist curve of this cylinder. By construction it is independent from the previous ones, and it does not intersect them. To complete the construction of the basis we have to chose a cycle B_{g} dual to A_{g}. Consider the following curve representing B_{g}:

— On any cylinder it follows the direction transversal to γ.

— On the surface S′_{i} the curve starts at the boundary component γ′′_{i}, then follows in the clockwise direction an arc in a small neighborhood of z_{i}, and finally arrives at the boundary component γ′_{i} in direction transversal to γ.

The canonical basis of cycles of S is constructed. By construction of this basis the index of any basic curve A_{i}, B_{i}, i = 1, …, g − 1 in the flat structure S does not depend on the way in which we applied the prescribed combination of figure eight constructions. Thus we have

∑_{k=1}^{g−1}(ind(A_{k}) + 1)(ind(B_{k}) + 1) = ∑_{i=1}^{p} ∑_{j_{i}}(ind(A_{j_{i}}) + 1)(ind(B_{j_{i}}) + 1) = ∑_{i=1}^{p} ϕ(S_{i})

Recall that the cycle A_{g} is represented by the waist curve of a cylinder. The tangent vector to the waist curve of the cylinder is constant in the flat structure ω. Thus ind(A_{g}) = 0, which implies that

ϕ(S) = ∑_{k=1}^{g}(ind(A_{k}) + 1)(ind(B_{k}) + 1) =

= 1 · (ind(B_{g}) + 1) + ∑_{k=1}^{g−1}(ind(A_{k}) + 1)(ind(B_{k}) + 1) =

= ind(B_{g}) + 1 + ∑_{i=1}^{p} ϕ(S_{i}) (mod 2)

It remains to compute the index ind(B_{g}). By the construction of B_{g}, we may assume that the tangent vector to the corresponding curve does not turn in the flat structure ω while it crosses the cylinders. It makes a turn by the angle 2π(a′_{i} + 1) while it follows a small arc joining two holes of a “figure eight” at the point z_{i} = w_{i}. This shows that ind(B_{g}) = ∑(a′_{i} + 1) (mod 2). □

Lemma 12.3. Suppose α is even so S has a spin structure. If there is some S′_{k} to which we apply the parallelogram construction and at least one of the points z_{k}, w_{k} is a zero (marked point) of even order, then all α′_{i} are even, and the parity of the spin structure of the flat surface S is equal to the sum of the parities of the spin structures of the components

ϕ(S) = ∑_{i=1}^{p} ϕ(S_{i})

Proof. We use the canonical basis on S similar to the one used in the Lemma above. To construct the first g − 1 pairs of curves we again use the curves living on the component surfaces S′_{i}. In addition to previous assumptions we require that when we apply parallelogram construction to some S_{i}, the basic curves do not approach the transversal ρ_{i} chosen to join z_{i} to w_{i}. We extend the description of the curve representing the cycle B_{g} to the surfaces to which we apply the parallelogram construction as follows: on such surfaces S′_{i} the curve starts at the boundary component γ′′_{i} in a direction transversal to γ and arrives at the boundary component γ′_{i} in direction transversal to γ. We chose this curve in such way that it does not intersect any basic curves on S′_{i}.

It follows from Lemma 12.1 that all the zeroes are of type III; we do not have any glued in cylinders: all the gluings are direct. As a curve representing the cycle A_{g} we now use a small smooth deformation γ̃ ⊂ S′_{k} of the boundary curve γ′_{k}, such that γ̃ does not pass through any zeroes of ω′_{k}. We get ind(A_{g}) = ind(γ̃) = (b′_{k} +1). Since the number b′_{k} is even, we see that ind(A_{g})+1 = 0 (mod 2). Thus the impact of the last pair of cycles to the sum

ϕ(S) = ∑_{k=1}^{g}(ind(A_{k}) + 1)(ind(B_{k}) + 1)

equals to zero, and

ϕ(S) = ∑_{k=1}^{g−1}(ind(A_{k}) + 1)(ind(B_{k}) + 1) =

= ∑_{i=1}^{p} ∑_{j_{i}}(ind(A_{j_{i}}) + 1)(ind(B_{j_{i}}) + 1) = ∑_{i=1}^{p} ϕ(S_{i})

Lemma 12.3 is proved. □

Note that there is a discrete freedom left in the figure eight or in the parallelogram construction. When we perform a figure eight construction at a zero of order a = a′ + a′′ with fixed a′, a′′ there are actually a + 1 ways to perform a figure eight construction, (see section 10.1). Similarly, when we perform a parallelogram construction at a pair of zeroes of orders b′, b′′ we, actually obtain (b′_{k} + 1) · (b′′_{k} + 1) different surfaces. Thus, even when we fix all the elements of the collection

(S′, J, a′_{j}, a′′_{j}, b′_{k}, b′′_{k})

we usually get numerous surfaces S defined by such assignment though some freedom may be cancelled by the symmetry. The collection α of degrees of zeros of the resulting flat structure S is, of course, invariant, but a spin structure may vary.

Lemma 12.4. Let N be the total number of surfaces S ∈ H(α) obtained by the assignment

(S′, J, a′_{j}, a′′_{j}, b′_{k}, b′′_{k}, γ, ρ, h, t) → S

with fixed (S′, J, a′_{j}, a′′_{j}, b′_{k}, b′′_{k}, γ, ρ, h, t). Suppose α is even so every such S has a spin structure. If there is some S′_{k} to which we apply the parallelogram construction, and one of the points z_{k}, w_{k} is an odd order zero, then exactly N/2 of the resulting surfaces have even spin structure and N/2 have odd spin structure.

Proof. Note that if one of b′_{k}, b′′_{k} is odd, the other one is odd as well.

Consider a canonical basis of cycles similar to one constructed in Lemma 12.3. The only difference is that now we may have some cylinders glued in. As in Lemma 12.2 we assume that the curve representing the cycle B_{g} stays transversal to the direction of γ while passing through the cylinders.

By the choice of this canonical basis of cycles the index of any basic curve representing A_{i}, B_{i}, i = 1, …, g − 1 in the flat structure S does not depend on the way in which we applied the prescribed combination of figure eight and parallelogram constructions. Thus the number

ϕ_{0} := ∑_{i=1}^{g−1}(ind(A_{i}) + 1)(ind(B_{i}) + 1)

is invariant of the discrete freedom in the assignment

(S′, J, a′_{j}, a′′_{j}, b′_{k}, b′′_{k}, γ, ρ, h, t) → S

We construct a representative of the cycle A_{g} as in Lemma 12.3 using a small smooth deformation γ̃ ⊂ S′_{k} of the curve γ′_{k}. We get ind(A_{g}) = ind(γ̃) = (b′_{k} + 1). Since the number b′_{k} is odd, we see that ind(γ̃) + 1 = 1 (mod 2). Thus the parity of the spin structure of S computed in the constructed canonical basis is represented as follows:

(24)  ϕ(S) = ϕ_{0} + (ind(A_{g}) + 1)(ind(B_{g}) + 1) = ϕ_{0} + ind(B_{g}) + 1 (mod 2)

Let z_{i} ≠ w_{i} be odd zero of ω′_{i} to which we apply parallelogram construction. Since the conical angle at z_{i} is 2(b′′_{i} + 1)π, there are an even number r = b′′_{i} + 1 of distinct points P_{1}, …, P_{r} of the form z_{i} + γ. We number the points in the natural cyclic order. Fix all the other elements of the construction leaving the only freedom in the choice of the point P_{j} to perform the slit z_{i}P_{j}. We get r = b′′_{i} + 1 surfaces S. We claim that r/2 of them have even parity of the spin structure, while another r/2 have odd parity of the spin structure.

To see this compare the surfaces obtained from the slits along z_{i}P_{j} and along z_{i}P_{j+1}. The two surfaces S share the same collection of curves representing the cycles A_{i}, B_{i}, i = 1, …, g − 1, and A_{g}. The curves representing the cycle B_{g} differ only near the point z_{i}: the tangent vector to one of the curves makes an extra turn by the angle 2π. Thus by equation (24) these two flat surfaces have opposite parities of the spin structures. □

12.2. Decomposition of Surfaces from Hyperelliptic Components. In this section we consider possible decompositions of flat surfaces S from hyperelliptic components corresponding to possible configurations of homologous saddle connections joining a zero to itself. In the next section we compute the constants in the quadratic asymptotics for the number of configurations of each type.

We start with the stratum H_{1}^{hyp}(α).

Lemma 12.5. Surfaces S in the hyperelliptic component H_{1}^{hyp}(α), g ≥ 3 are obtained by an assignment

(S′, J, a′_{j}, a′′_{j}, b′_{k}, b′′_{k}, γ, ρ, h, t) → S

of one of the following three types: i) (\overline{g − 2}, \overline{g − 2}) →

The multiplicity is 1; the flat surface S′ belongs to the hyperelliptic component H_{1}^{hyp}(g−2, g−2); we apply the parallelogram construction gluing the boundaries directly; the points z+γ, w+γ in the parallelogram construction are chosen to be symmetric with respect to the hyperelliptic involution on S′. For fixed data there precisely g − 1 surfaces S that can be built.

ii) (\overline{(g − 2) + (g − 2)}) ⇒

The multiplicity is 1; the flat surface S′ belongs to the hyperelliptic component H_{1}^{hyp}(2g − 4); we apply the figure eight construction gluing in a cylinder; a′ = a′′ = g − 2. There are 2g − 3 surfaces S that can be built.

iii) (\overline{(g_{1} − 1) + (g_{1} − 1)}) → (\overline{g_{2} − 1}, \overline{g_{2} − 1}) →

The multiplicity is 2; the flat surfaces belong to the hyperelliptic components S′_{1} ∈ H^{hyp}(2g_{1} − 2), S′_{2} ∈ H^{hyp}(g_{2} − 1, g_{2} − 1), where g_{1} + g_{2} = g − 1. We apply the figure eight construction with a′_{1} = a′′_{1} = g_{1} − 1 to the first surface and the parallelogram construction with b′_{2} = b′′_{2} = g_{2} − 1 to the second one gluing both pairs of components directly. The points z_{2} + γ, w_{2} + γ ∈ S′_{2} are chosen to be symmetric with respect to the hyperelliptic involution on S′_{2}. There are g_{1}(2g_{2} − 1) surfaces that can be built.

The surfaces can be related by a stratum interchange or by a γ → −γ symmetry.

The proof is analogous to the proof of Lemma 8.3. Decompositions of the flat surfaces from the hyperelliptic connected component

H_{1}^{hyp}(g − 1, g − 1) are described by the following Lemma.

Lemma 12.6. Surfaces S in the hyperelliptic component H_{1}^{hyp}(g − 1, g − 1) are obtained by an assignment

(S′, J, a′_{j}, a′′_{j}, b′_{k}, b′′_{k}, γ, ρ, h, t) → S

of one of the following two types.

i) (\overline{g − 2}, \overline{g − 2}) ⇒

The multiplicity is 1; S′ ∈ H_{1}^{hyp}(g−2, g−2). We apply the parallelogram construction gluing in a cylinder; b′_{1} = b′′_{1} = g − 2. The points z + γ, w + γ in the parallelogram construction are chosen to be symmetric with respect to the hyperelliptic involution on S′. For fixed data there are precisely g − 1 surfaces S that can be built.

ii) (\overline{(g_{1} − 1)}, \overline{(g_{1} − 1)}) → (\overline{g_{2} − 1}, \overline{g_{2} − 1}) →

The multiplicity is 2; S′_{i} ∈ H^{hyp}(g_{i} − 1, g_{i} − 1). where g_{1} + g_{2} = g − 1. We apply two parallelogram constructions gluing the components directly. The points z_{i} +γ, w_{i} +γ in the parallelogram constructions are chosen to be symmetric with respect to the hyperelliptic involution on each S′_{i}, i = 1, 2. For the fixed data the number of surfaces S that can be built is g_{1} · g_{2} for g_{1} ≠ g_{2} and g_{1} · g_{2}/2 for g_{1} = g_{2}.

Surfaces can be related by a stratum interchange or by a γ → −γ symmetry.

The proof is analogous to the proof of Lemma 8.3.

12.3. Constants for the Hyperelliptic Connected Components. The computation of the constants is based essentially on (23). However, since the flat surfaces in the hyperelliptic components have extra symmetry we have to make necessary adjustments.

12.3.1. Hyperelliptic Component H_{1}^{hyp}(2g −2). The admissible assignments for this component are described in Lemma 12.5. We always have the γ → −γ symmetry for all three assignments. We never have any stratum interchange symmetry.

The only modification which we need to make in equation (23) is as follows. Applying the parallelogram constructions in this case we need to chose the points z_{i} + γ, w_{i} + γ to be symmetric with respect to hyperelliptic involution on each S_{i} (see Lemma 12.5). Note that for this component we always have b′_{k} = b′′_{k} in any assignment containing a parallelogram construction. Thus we have to replace each factor (b′_{k} + 1)(b′′_{k} + 1) by the corresponding factor (b′_{k} + 1). We get the following

Formula 12.1. For almost all flat surfaces in a hyperelliptic connected component H_{1}^{hyp}(α), g ≥ 3, there are only three possible types of configurations of saddle connections joining zero to itself. The constants in the quadratic asymptotics for the number of saddle connections of each type are presented by the following formulae:

i) Assignment (\overline{g − 2}, \overline{g − 2}) →

c = \frac{g − 1}{2} · \frac{Vol(H_{1}^{hyp}(g − 2, g − 2))}{Vol(H_{1}^{hyp}(2g − 2))}

ii) Assignment (\overline{(g − 2) + (g − 2)}) ⇒

c = \frac{1}{2} · \frac{2g − 3}{2g − 2} · \frac{Vol(H_{1}^{hyp}(2g − 4))}{Vol(H_{1}^{hyp}(2g − 2))}

iii) Assignment (\overline{(g_{1} − 1) + (g_{1} − 1)}) → (\overline{g_{2} − 1}, \overline{g_{2} − 1}) →

c = \frac{(2g_{1} − 1)g_{2}}{4} · \frac{(2g_{1} − 1)! (2g_{2})!}{(2g − 2)!} · \frac{Vol(H_{1}^{hyp}(2g_{1} − 2)) Vol(H_{1}^{hyp}(g_{2} − 1, g_{2} − 1))}{Vol(H_{1}^{hyp}(2g − 2))}

where g_{1} + g_{2} = g − 1.

Example 12.7. Stratum H(2). This stratum is connected; it coincides with its hyperelliptic component. Flat surfaces from this stratum have a single zero of order 2. It is easy to see that in genus g = 2 only multiplicity one is realizable.

→ (\overline{0}, \overline{0}) → In this case is a single saddle connection returning with angle 3π so that there is no cylinder. After collapsing this saddle connection we get a torus with two marked points to which we apply the parallelogram construction, gluing the circles directly. Here we have

c = \frac{2 − 1}{2} · \frac{Vol(H_{1}(∅))}{Vol(H_{1}(2))} = \frac{10}{3} · \frac{1}{ζ(2)} ≈ 2.026

⇒ (\overline{0}) ⇒ The other possibility is that the saddle connection returns at angle π. Then there is a whole cylinder filled with homologous closed geodesics. In particular, there is another saddle connection returning at the angle π homologous to the initial one. After collapsing the cylinder we get a torus with a single marked point. We apply the figure eight construction.

c = \frac{1}{4} · \frac{Vol(H_{1}(∅))}{Vol(H_{1}(2))} = \frac{5}{3} · \frac{1}{ζ(2)} ≈ 1.013

Example 12.8. Component H^{hyp}(4). In genus g = 3 all three possible assignments (see Lemma 12.5) are already admissible.

→ (\overline{1}, \overline{1}) → The first represents the multiplicity 1 case of a saddle connection returning at angle 5π.

c = \frac{3 − 1}{2} · \frac{Vol(H(1, 1))}{Vol(H^{hyp}(4))} = \frac{224}{27} · \frac{1}{ζ(2)} ≈ 5.044

⇒ (\overline{1 + 1}) ⇒ If there is a cylinder, by symmetry, the spacing of the angles between the boundaries of the cylinders is 4π. Then after degeneration, we get a single zero of order 2 on a surface of genus 2. We apply the figure eight construction with a′_{1} = a′′_{1} = 1, and glue in a cylinder. We get

c = \frac{1}{2} · \frac{3}{4} · \frac{Vol(H(2))}{Vol(H^{hyp}(4))} = \frac{7}{2} · \frac{1}{ζ(2)} ≈ 2.128

→ (\overline{0 + 0}) → (\overline{0}, \overline{0}) → In the multiplicity p = 2 case we have two homologous curves, both returning to z_{0} at angles 3π. After degeneration we get two tori, one with two marked points, the other with one. We perform the figure eight construction on one torus and the parallelogram construction on the other, gluing the boundaries directly to each other.

c = \frac{(2 · 1 − 1) · 1}{4} · \frac{(2 · 1 − 1)! (2 · 1)!}{(6 − 1)!} · \frac{Vol(H(∅))^{2}}{Vol(H^{hyp}(4))} = \frac{70}{27} · \frac{1}{ζ(2)} ≈ 1.576

12.3.2. Hyperelliptic Component H^{hyp}(g − 1, g − 1). The admissible assignments for this component are described by Lemma 12.6. We always have the γ → −γ symmetry for both assignments. We have the stratum interchange symmetry in the second assignment if and only if g_{1} = g_{2}.

The only modification which we need to make in the equation (23) is as follows. Applying the parallelogram constructions in this case we need to chose the points z_{i}+γ, w_{i}+γ to be symmetric with respect to hyperelliptic involution on each (S_{i}, ω_{i}) (see Lemma 12.6). This means that we have to replace each factor (b′_{k} + 1)(b′′_{k} + 1) by the corresponding factor (b′_{k} + 1). Thus we get the following

Formula 12.2. For almost all flat surfaces in the hyperelliptic connected component H^{hyp}(g−1, g−1), g ≥ 3, there are only two possible types of configurations of saddle connections joining any of two zeroes to itself. The constants in the quadratic asymptotics for the number of saddle connections of each type are presented by the following formulae:

i) Assignment (\overline{g − 2}, \overline{g − 2}) ⇒

c = \frac{(g − 1)}{(2g − 1)} · \frac{Vol(H_{1}^{hyp}(g − 2, g − 2))}{Vol(H_{1}^{hyp}(g − 1, g − 1))}

ii) Assignment (\overline{(g_{1} − 1)}, \overline{(g_{1} − 1)}) → (\overline{g_{2} − 1}, \overline{g_{2} − 1}) →

c = \frac{g_{1}g_{2}}{2|Γ|} · \frac{(2g_{1})! (2g_{2})!}{(2g − 1)!} · \frac{Vol(H_{1}^{hyp}(g_{1} − 1, g_{1} − 1)) Vol(H_{1}^{hyp}(g_{2} − 1, g_{2} − 1))}{Vol(H_{1}^{hyp}(g − 1, g − 1))}.

where g_{1} + g_{2} = g − 1, and

|Γ| = \begin{cases}1 & when g_{1} ≠ g_{2} \\ 2 & when g_{1} = g_{2} \end{cases}

Example 12.9. Stratum H(1, 1). In genus g = 2 the stratum H(1, 1) is the principal stratum. It is connected, and it coincides with the hyperelliptic component H(1, 1) = H^{hyp}(1, 1).

Since g = g_{1} + g_{2} + 1, and g_{i} ≥ 1, we see that multiplicity two does not occur in genus g = 2. The value

c = \frac{1}{3} · \frac{Vol(H_{1}^{hyp}(∅))}{Vol(H_{1}^{hyp}(1, 1))} = \frac{5}{2}

of the constant in multiplicity one was computed in section 11.3.

Example 12.10. Component H^{hyp}(2, 2).

⇒ (\overline{1}, \overline{1}) ⇒ In the multiplicity 1 case there is a cylinder returning to the other zero. Here we get

c = \frac{2}{5} · \frac{Vol(H_{1}(1, 1))}{Vol(H_{1}^{hyp}(2, 2))} = \frac{14}{3} · \frac{1}{ζ(2)} ≈ 2.837

→ (\overline{0}, \overline{0}) → (\overline{0}, \overline{0}) → In the multiplicity 2 case we have homologous curves, neither of which bounds a cylinder, one returning to each zero. The degenerating surfaces are tori each with 2 marked points on each. We perform the parallelogram construction, gluing the circles pairwise directly. Since g_{1} = g_{2} we get not only the γ → −γ but also the stratum interchange, |Γ| = 2.

c = \frac{1 · 1}{2 · 2} · \frac{(2 · 1)! (2 · 1)!}{(2 · 3 − 1)!} · \frac{Vol(H_{1}(∅))^{2}}{Vol(H_{1}^{hyp}(2, 2))} = \frac{35}{24} · \frac{1}{ζ(2)} ≈ 0.8866

12.4. Connected Component H^{nonhyp}(g − 1, g − 1); Even Genus g. When genus g ≥ 4 is even, the stratum H(g −1, g −1) has two connected components: the hyperelliptic one, H^{hyp}(g−1, g−1), which we considered in the previous section, and the nonhyperelliptic connected component H^{nonhyp}(g − 1, g − 1). The description of admissible constructions for the connected component H^{nonhyp}(g − 1, g − 1) for even g ≥ 4 is completely analogous to the case of connected strata. However, in the following two assignments

(\overline{g − 2}, \overline{g − 2}) ⇒

and

(\overline{(g_{1} − 1)}, \overline{(g_{1} − 1)}) → (\overline{g_{2} − 1}, \overline{g_{2} − 1}) →

we have interference with the hyperelliptic components. Thus in these two cases we have to modify the general formula 11.1 by subtracting the correction term corresponding to the constructions leading to hyperelliptic flat surfaces, see formula 12.2.

Formula 12.3. For almost all flat surfaces in a nonhyperelliptic connected component H^{nonhyp}(g−1, g−1) for even g ≥ 4, the constants in the quadratic asymptotics for the number of saddle connections of each of the following two types are presented by the following formulae:

i) Assignment (\overline{g − 2}, \overline{g − 2}) ⇒

c = \frac{(g − 1)^{2}}{(2g − 1)} · \frac{Vol(H_{1}(g − 2, g − 2))}{Vol(H_{1}^{nonhyp}(g − 1, g − 1))} − \frac{(g − 1)}{(2g − 1)} · \frac{Vol(H_{1}^{hyp}(g − 2, g − 2))}{Vol(H_{1}^{nonhyp}(g − 1, g − 1))}

ii) Assignment (\overline{(g_{1} − 1)}, \overline{(g_{1} − 1)}) → (\overline{g_{2} − 1}, \overline{g_{2} − 1}) →

c = \frac{(g_{1})^{2}(g_{2})^{2}}{2|Γ|} · \frac{(2g_{1})! (2g_{2})!}{(2g − 1)!} · \frac{Vol(H_{1}(g_{1} − 1, g_{1} − 1)) Vol(H_{1}(g_{2} − 1, g_{2} − 1))}{Vol(H_{1}^{nonhyp}(g − 1, g − 1))} −

− \frac{g_{1}g_{2}}{2|Γ|} · \frac{(2g_{1})! (2g_{2})!}{(2g − 1)!} · \frac{Vol(H_{1}^{hyp}(g_{1} − 1, g_{1} − 1)) Vol(H_{1}^{hyp}(g_{2} − 1, g_{2} − 1))}{Vol(H_{1}^{nonhyp}(g − 1, g − 1))}

where g_{1} + g_{2} = g − 1, and

|Γ| = \begin{cases}1 & when g_{1} ≠ g_{2} \\ 2 & when g_{1} = g_{2} \end{cases}

iii) The constants for all other assignments are given by equation (23), with α = (g − 1, g − 1) where Vol(H(g − 1, g − 1)) in the denominator of the rightmost fraction should be replaced by Vol(H_{1}^{nonhyp}(g − 1, g − 1)).

As an example of this formula we present the list of possible generic degenerations of a flat surface S ∈ H^{nonhyp}(3, 3) in Appendix B.

Remark 12.11. Note, that one can represent the constant corresponding to the two exceptional assignments in the formula above as

c = c_{regular} − c_{hyp} · \frac{Vol(H^{hyp}(g − 1, g − 1))}{Vol(H^{nonhyp}(g − 1, g − 1))}.

Here c_{hyp} in the correctional term is the constant for the corresponding assignment for the hyperelliptic component H^{hyp}(g − 1, g − 1), see Formula 12.2.

12.5. Nonhyperelliptic Components with a Spin Structure. Let now α be even and let S belong to one of the nonhyperelliptic components H_{1}^{even}(α) or H^{odd}(α). The admissible assignments for such flat surfaces are described by the general Lemmas 12.2, 12.3, 12.4.

When α is one of the two partitions (2g − 2), (g − 1, g − 1) there are special assignments where there might be interference with hyperelliptic components, see Lemmas 12.5, and 12.6.

In the formula below we use function δ(α′, ϕ′) which is equal to 1 when the stratum H(α′) contains a hyperelliptic component, and when, moreover, this hyperelliptic component has parity ϕ′ of the spin structure. This function is equal to zero otherwise (see (16) for the explicit definition). By convention we let the volume of several “missing” components in small genera and the volume of nonexisting hyperelliptic components be equal to zero (see (17) for the complete list).

Formula 12.4. Let all the integers in α be even, and let α be different from any of the partitions (2g − 2), (g − 1, g − 1).

For almost all flat surfaces in the connected component H^{ϕ}(α) the constants in the quadratic asymptotics for the number of saddle connections joining some zero to itself are presented by the following formulae:

MODULI OF ABELIAN DIFFERENTIALS: THE PRINCIPAL BOUNDARY 71

— If the assignment contains only figure eight constructions

c = \frac{1}{|Γ_{−}| · |Γ|} · ∏_{m∈α} (\frac{o(m)!}{∏_{i=1}^{p} o_{i}(m)!}) · ∏_{\begin{matrix}1≤i≤p \\ a_{i}∈α \end{matrix}} o_{i}(a_{i}) · ∏_{1≤i≤p}(a_{i} + 1)

· \frac{1}{2^{p−1}} · \frac{∏_{i=1}^{p}(\frac{d_{i}}{2} − 1)!}{(\frac{d}{2} − 2)!} · \frac{1}{Vol(H_{1}^{ϕ}(α))}·

· ∑_{\begin{matrix}ϕ′_{1},...,ϕ′_{p}∈\{even, odd\} \\ (ϕ′_{1}+···+ϕ′_{p})+ \\ +(a′_{1}+ ˙+a′_{p})+ \\ +p+1≡ϕ (mod 2) \end{matrix}} ∏_{i=1}^{p} ( Vol(H_{1}^{ϕ_{i}}(α′_{i})) + δ(α′_{i}, ϕ′_{i}) Vol(H_{1}^{hyp}(α′_{i})))

— If the assignment contains at least one parallelogram construction with even b′_{i}

c = \frac{1}{|Γ_{−}|} · \frac{1}{|Γ|} · ∏_{m∈α} (\frac{o(m)!}{∏_{i=1}^{p} o_{i}(m)!}) ·

· ∏_{\begin{matrix}1≤i≤p \\ z_{i}=w_{i} \\ a_{i}∈α \end{matrix}} o_{i}(a_{i}) · ∏_{\begin{matrix}1≤k≤p \\ z_{k}≠w_{k} \\ b′_{k}≠b′′_{k} \\ b′_{k}∈α \end{matrix}} o_{k}(b′_{k}) · ∏_{\begin{matrix}1≤k≤p \\ z_{k}≠w_{k} \\ b′_{k}≠b′′_{k} \\ b′′_{k}∈α \end{matrix}} o_{k}(b′′_{k}) · ∏_{\begin{matrix}1≤k≤p \\ z_{k}≠w_{k} \\ b′_{k}=b′′_{k}∈α \end{matrix}} o_{k}(b′_{k})(o_{k}(b′_{k}) − 1)·

· ∏_{\begin{matrix}1≤i≤p \\ z_{i}=w_{i} \end{matrix}}(a_{i} + 1) · ∏_{\begin{matrix}1≤k≤p \\ z_{k}≠w_{k} \end{matrix}}(b′_{k} + 1)(b′′_{k} + 1) · \frac{1}{2^{p−1}} · \frac{∏_{i=1}^{p}(\frac{d_{i}}{2} − 1)!}{(\frac{d}{2} − 2)!}·

· \frac{1}{Vol(H_{1}^{ϕ}(α))} · ∑_{\begin{matrix}ϕ′_{1},...,ϕ′_{p}∈\{even, odd\} \\ (ϕ′_{1}+···+ϕ′_{p})≡ϕ (mod 2) \end{matrix}} ∏_{i=1}^{p} ( Vol(H_{1}^{ϕ_{i}}(α′_{i})) + δ(α′_{i}, ϕ′_{i}) Vol(H_{1}^{hyp}(α′_{i})))

— If the assignment contains at least one parallelogram construction with odd b′_{i}

c = \frac{1}{|Γ_{−}|} · \frac{1}{|Γ|} · ∏_{m∈α} (\frac{o(m)!}{∏_{i=1}^{p} o_{i}(m)!}) ·

· ∏_{\begin{matrix}1≤i≤p \\ z_{i}=w_{i} \\ a_{i}∈α \end{matrix}} o_{i}(a_{i}) · ∏_{\begin{matrix}1≤k≤p \\ z_{k}≠w_{k} \\ b′_{k}≠b′′_{k} \\ b′_{k}∈α \end{matrix}} o_{k}(b′_{k}) · ∏_{\begin{matrix}1≤k≤p \\ z_{k}≠w_{k} \\ b′_{k}≠b′′_{k} \\ b′′_{k}∈α \end{matrix}} o_{k}(b′′_{k}) · ∏_{\begin{matrix}1≤k≤p \\ z_{k}≠w_{k} \\ b′_{k}=b′′_{k}∈α \end{matrix}} o_{k}(b′_{k})(o_{k}(b′_{k}) − 1)·

· ∏_{\begin{matrix}1≤i≤p \\ z_{i}=w_{i} \end{matrix}}(a_{i} + 1) · ∏_{\begin{matrix}1≤k≤p \\ z_{k}≠w_{k} \end{matrix}}(b′_{k} + 1)(b′′_{k} + 1)·

· \frac{1}{2} · \frac{1}{2^{p−1}} · \frac{∏_{i=1}^{p}(\frac{d_{i}}{2} − 1)!}{(\frac{d}{2} − 2)!} · \frac{∏_{i=1}^{p} Vol(H_{1}(α′_{i}))}{Vol(H_{1}^{ϕ}(α))}

Proof. The formula above is obtained by elementary adjustment of formula (23).

First suppose that the assignment under consideration does not involve any parallelogram constructions with odd zeroes. Then by Lemma 12.2 and Lemma 12.3 the spin structure of S is determined by the spin structures of S′_{i} and by the parities of a′_{i}. Thus the combinatorial constant is exactly the same as in equation (23). The numerator of the volume term is the sum over those products of Vol(H^{even}(α_{i})), Vol(H^{odd}(α_{j})), and Vol(H^{hyp}(α_{i})) (when appropriate) that satisfy the relevant relation between spin structures of component surfaces and the spin structure of the resulting surface, see Lemma 12.2 and Lemma 12.3 correspondingly. In the denominator of the volume term we have the volume of the component under consideration.

If there is a parallelogram construction with odd b′_{k} involved in the assignment, then by Lemma 12.4 we have to multiply c by 1/2 to account for the fact that half of the constructions produce flat surfaces with even spin structure and another half of the constructions produce a flat surface with odd spin structure. Thus we need to divide by 2 the combinatorial factor in equation (23). In this case the numerator of the volume term is same as in (23): it is a product of Vol(H_{1}(α_{i})), where we take the total volume of all connected components in the stratum. In the denominator of the volume term we again have the volume of the component under consideration. □

We complete this section considering the exceptional strata.

Formula 12.5. Let α = (2g − 2), or let α = (g − 1, g − 1) where g is odd. Let g ≥ 4, or let g = 3 and ϕ = odd.

For almost all flat surfaces in the connected component H^{ϕ}(α) the constants in the quadratic asymptotics for the number of saddle connections joining some zero to itself are presented by the following formulae:

— If the assignment is different from any of the assignments listed in Lemmas 12.5, and 12.6, then the constant is the same as in Formula 12.4.

— For any of the assignments listed in one of the Lemmas 12.5, and 12.6 the constant is equal to

c = c_{regular} − c_{hyp} · \frac{Vol(H^{hyp}(α))}{Vol(H^{ϕ}(α))}.

Here c_{regular} is given by Formula 12.4, and c_{hyp} is the constant for the corresponding assignment for the hyperelliptic component H^{hyp}(g−1, g−1), see Formulae 12.1 and 12.2.

Proof. For nonexceptional assignments the proof is exactly the same as for Formula 12.4. For the exceptional assignments, which may lead to the flat surfaces from hyperelliptic components one makes appropriate adjustment, see the proof of Formula 12.3 and Remark 12.11 after this proof. □

Example 12.12. Component H^{odd}(4). Recall that the stratum H(4) has only one nonhyperelliptic connected component H^{odd}(4). Every flat surface in H(4) having even parity of the spin structure belongs to H^{hyp}(4). We have d = dim_{ℝ} H(4) = 12. Consider possible assignments for the flat surfaces from the component H^{odd}(4). We start with assignments of multiplicity one.

→ (\overline{2}, \overline{0}) → We have a single saddle connection that does not bound a cylinder. It returns at angle 3π. After degeneration we get a surface of genus 2 with a zero of order 2 and with a marked point. We apply the parallelogram construction gluing the circles directly. Since b′_{1} = 2, and b′′_{1} = 0 are even, by lemma 12.3 the parity of the spin structure of the resulting surface coincides with the parity of the spin structure of a surface from H(2), so it is really odd, see (7). Since b′_{1} ≠ b′′_{2} we do not have a γ → −γ symmetry. Thus,

c = 3 · \frac{(5 − 1)!}{(6 − 2)!} · \frac{Vol(H(2))}{Vol(H^{odd}(4))} = \frac{81}{8} · \frac{1}{ζ(2)} ≈ 6.155

→ (\overline{1}, \overline{1}) → Here we again have a single saddle connection that does not bound a cylinder, but now it returns at angle 5π. After degeneration we get a surface of genus 2 with a pair of simple zeroes. We apply the parallelogram construction gluing the circles directly. Since b′_{1} = 1, and b′′_{1} = 1 we get N = (b′_{1} + 1)(b′′_{1} + 1) = 4 surfaces, but only half of them have odd parity of the spin structure, see lemma 12.4. Now we have a γ → −γ symmetry, |Γ_{−}| = 2.

c = 1 · \frac{(5 − 1)!}{(6 − 2)!} · \frac{Vol(H(1, 1))}{Vol(H^{odd}(4))} = 3 · \frac{1}{ζ(2)} ≈ 1.824

⇒ (\overline{0 + 2}) ⇒ Suppose now that the saddle connection returns at angle π. Then it bounds a cylinder filled with homologous closed geodesics. In particular, there is another saddle connection returning at the angle π homologous to the initial one. Since the surface is not hyperelliptic, the angle between the saddle connections on the opposite sides of the cylinder is 6π. After degeneration we get a single zero of order 2 on a surface of genus 2. We apply the figure eight construction, breaking the zero of order 2 into a zero of order 2 and one of order 0. This corresponds to spacings of 2π and 6π, or, in our notation to the choice a′_{1} = 0, a′′_{1} = 2. We glue in an intervening cylinder. By lemma 12.2 the parity of the spin structure of the resulting flat surface equals 1 + ϕ(S′_{1}) + (0 + 1) = 1 since ϕ(S′_{1}) = 1, see (7). Since a′_{1} ≠ a′′_{1} we do not have the γ → −γ symmetry.

c = 3 · \frac{(4 − 1)!}{(6 − 2)!} · \frac{Vol(H(2))}{Vol(H^{odd}(4))} = \frac{81}{32} · \frac{1}{ζ(2)} ≈ 1.539

⇒ (\overline{0 + 0}) → (\overline{0 + 0}) ⇒ We now consider multiplicity 2. We have a cylinder and a homologous curve that does not bound a cylinder. The degenerated surfaces S′_{1}, S′_{2} are a pair of tori, each with a marked point. We apply the figure eight construction giving a pair of circles on each torus. We glue one pair of circles directly and put an intervening cylinder between the other pair. Though α′_{1} = α′_{2} we do not have a stratum interchange, |Γ| = 1, but we have the γ → −γ symmetry.

c = \frac{1}{2} · \frac{1}{2^{2−1}} · \frac{(2 − 1)! · (2 − 1)!}{(6 − 2)!} · \frac{Vol(H(∅))^{2}}{Vol(H^{odd}(4))} = \frac{15}{32} · \frac{1}{ζ(2)} ≈ 0.2850

Example 12.13. Component H^{odd}(2, 2).

→ (2, \overline{0}, \overline{0}) → In the multiplicity 1 case with no cylinder, the degenerating surface is genus 2 with a pair of marked points and a zero of order 2. We perform the parallelogram construction on the marked points, gluing the circles directly. Since b′_{1} = b′′_{1} = 0 is even, by lemma 12.3 the parity of the spin structure equals ϕ(S′), which is odd for S′ ∈ H(2).

c = 1 · \frac{(4 + 2 − 1)!}{(7 − 2)!} · \frac{Vol(H_{1}(2))}{Vol(H_{1}^{odd}(2, 2))} = 6 · \frac{1}{ζ(2)} ≈ 3.648

⇒ (2, \overline{0 + 0}) ⇒ In the case of a cylinder with multiplicity 1, if the other side returns to the same zero, we get a genus 2 surface with a single marked point and a double zero. We perform the figure eight construction at the marked point, gluing in a cylinder to the two circles. We have a′_{1} = a′′_{1} = 0. Thus, by lemma 12.2 ϕ(S) = 1+ϕ(S′)+(0+1) = 1 + 1 + 1 = 1 mod 2. Now |Γ_{|} = 2.

c = 1 · \frac{(4 + 1 − 1)!}{(7 − 2)!} · \frac{Vol(H_{1}(2))}{Vol(H_{1}^{odd}(2, 2))} = \frac{6}{5} · \frac{1}{ζ(2)} ≈ 0.7295

⇒ (\overline{1}, \overline{1}) ⇒ If the other side of the cylinder returns to the other zero, after degeneration we get 2 simple zeroes. We perform the parallelogram construction gluing in a cylinder. Now b′_{1} = b′′_{1} = 1 is odd. Thus we get N = (b′_{1} + 1)(b′′_{1} + 1) = 4 surfaces, but by lemma 12.4 only N/2 = 2 of them have odd parity of the spin structure. There is also the γ → −γ symmetry.

c = 2 · \frac{(5 − 1)!}{(7 − 2)!} · \frac{Vol(H_{1}(1, 1))}{Vol(H_{1}^{odd}(2, 2))} = \frac{32}{15} · \frac{1}{ζ(2)} ≈ 1.297

⇒ (\overline{0 + 0}) ⇒ (\overline{0 + 0}) ⇒ In the multiplicity 2 case we have a pair of cylinders. The two sides of each cylinder return to different zeroes. The result is a pair of tori, each with a single marked point. We perform the figure eight construction creating two pairs of circles to which we glue intervening cylinders. We have |Γ| = |Γ_{−}| = 2.

c = \frac{1}{2} · \frac{1}{2^{2−1}} · \frac{(2 − 1)! (2 − 1)!}{(7 − 2)!} · \frac{Vol(H_{1}(∅)) Vol(H_{1}(∅))}{Vol(H_{1}^{odd}(2, 2))} = \frac{1}{6} · \frac{1}{ζ(2)} ≈ 0.1013

Acknowledgments

The third author wants to thank MPI für Mathematik at Bonn and IHES at Bures-sur-Yvette for their hospitality while preparation of this paper.

Appendix. Values of the Constants

Appendix A. Saddle Connections Joining Distinct Zeroes

A.1. Connected Strata in Genus 4. In the tables below we consider all connected strata in genus g = 4. For each stratum we present the complete list of all possible topological configurations of geodesic saddle connections. To make calculations traceable we present for every configuration the orders of possible symmetries Γ and Γ_{−}, the combinatorial constant M, and the result: the Siegel–Veech constant in quadratic asymptotics for the number of saddle connections (closed geodesics) of that type. The notations were introduced in Section 7.2 and Section 10.3

Stratum H(5, 1).

Degeneration pattern  |Γ_{−}| |Γ| M  c  c approx.
(1 + 5) ≻  1  1  7 \frac{4311167}{373248} 11.5504
(0 + 0) ≻ (0 + 4) ≻  1  1  5  \frac{38125}{93312}  0.408576
(0 + 2) ≻ (0 + 2) ≻  1  2  \frac{9}{2}  \frac{21}{512}  0.0410156

Stratum H(4, 1, 1).

(1 + 1, 4) ≻  2  1  3  \frac{2403}{616}  3.90097
(1 + 4, 1) ≻  1  1 12 \frac{186624}{9625}  19.3895
(0 + 0) ≻ (0 + 0, 4) ≻  2  1  1  \frac{61}{616}  0.099026
(0 + 0) ≻ (0 + 3, 1) ≻  1  1  8  \frac{1024}{1925}  0.531948
(0 + 2) ≻ (0 + 1, 1) ≻  1  1 12  \frac{108}{1375}  0.0785455

Stratum H(3, 2, 1).

(1 + 2, 3) ≻  1  1  4  \frac{368}{63}  5.84127
(1 + 3, 2) ≻  1  1  5  \frac{55625}{7168}  7.76018
(2 + 3, 1) ≻  1  1  6  \frac{81}{7}  11.5714
(0 + 0) ≻ (0 + 2, 2) ≻  1  1  3  \frac{765}{3584}  0.213449
(0 + 0) ≻ (0 + 1, 3) ≻  1  1  2  \frac{10}{63}  0.15873
(0 + 0) ≻ (1 + 2, 1) ≻  1  1  4  \frac{20}{63}  0.31746
(0 + 0, 2) ≻ (0 + 2) ≻  1  1  3  \frac{27}{1024}  0.0263672
(0 + 2) ≻ (1 + 0, 1) ≻  1  1  6  \frac{3}{64}  0.046875
(1 + 1) ≻ (0 + 1, 1) ≻  1  1  6  \frac{3}{64}  0.046875
(0 + 0) ≻ (0 + 0) ≻ (0 + 1, 1) ≻  1  1  2  \frac{5}{288}  0.0173611

76 ALEX ESKIN, HOWARD MASUR, AND ANTON ZORICH

Configurations of geodesic saddle connections.

Stratum H(3, 1, 1, 1).

Degeneration pattern  |Γ_{−}| |Γ| M  c  c approx.
(1 + 1, 3, 1) ≻  2  1  9  \frac{729}{62}  11.7581
(1 + 3, 1, 1) ≻  1  1 15 \frac{185625}{7936}  23.3902
(0 + 0) ≻ (0 + 0, 3, 1) ≻  2  1  3  \frac{15}{62}  0.241935
(0 + 0) ≻ (0 + 2, 1, 1) ≻  1  1  9  \frac{2025}{3968}  0.510333
(0 + 2) ≻ (0 + 0, 1, 1) ≻  1  1  9  \frac{405}{7936}  0.0510333
(0 + 1, 1) ≻ (0 + 1, 1) ≻  1  2 12  \frac{3}{62}  0.0483871

Stratum H(2, 2, 1, 1).

(1 + 1, 2, 2) ≻  2  1  3  \frac{4101}{1048}  3.91317
(1 + 2, 2, 1) ≻  1  1 16  \frac{3072}{131}  23.4504
(2 + 2, 1, 1) ≻  2  1  5  \frac{6875}{786}  8.74682
(0 + 0) ≻ (0 + 0, 2, 2) ≻  2  1  1  \frac{85}{1048}  0.0811069
(0 + 0) ≻ (0 + 1, 2, 1) ≻  1  1  8  \frac{200}{393}  0.508906
(0 + 0) ≻ (1 + 1, 1, 1) ≻  2  1  3  \frac{25}{131}  0.19084
(0 + 0, 2) ≻ (0 + 0, 2) ≻  2  2  1  \frac{3}{524}  0.00572519
(0 + 0, 2) ≻ (0 + 1, 1) ≻  1  1  8  \frac{16}{393}  0.0407125
(1 + 1) ≻ (0 + 0, 1, 1) ≻  2  1  3  \frac{5}{262}  0.019084
(0 + 1, 1) ≻ (1 + 0, 1) ≻  2  1  8  \frac{128}{3537}  0.0361889
(0 + 0) ≻ (0 + 0) ≻ (0 + 0, 1, 1) ≻  2  1  1  \frac{25}{3537} 0.00706814

Stratum H(2, 1, 1, 1, 1).

(1 + 1, 2, 1, 1) ≻  2  1 18  \frac{1179}{50}  23.58
(1 + 2, 1, 1, 1) ≻  1  1 16 \frac{15872}{675}  23.5141
(0 + 0) ≻ (0 + 0, 2, 1, 1) ≻  2  1  6  \frac{2}{5}  0.4
(0 + 0) ≻ (0 + 1, 1, 1, 1) ≻  1  1  8  \frac{56}{135}  0.414815
(0 + 0, 2) ≻ (0 + 0, 1, 1) ≻  2  1  6  \frac{1}{50}  0.02
(0 + 0, 1, 1) ≻ (0 + 1, 1) ≻  1  1 24  \frac{16}{225}  0.0711111

Table 8. Principal stratum H(1, …, 1). Approximate values of the constants c for saddle connections of multiplicity two joining a pair of distinct zeroes.

H(\overbrace{1, …, 1}^{2g_{1}−2}, 0 + 0) ≻ H(\overbrace{1, …, 1}^{2g_{2}−2}, 0 + 0) ≻ g_{1} + g_{2} = g

g_{2} = 1  g_{2} = 2  g_{2} = 3  g_{2} = 4  g_{2} = 5  g_{2} = 6
g_{1} = 1  0.6250  0.8571  0.8355  0.8393  0.8426  0.8435
g_{1} = 2  0.8571  0.07958  0.07193  0.04160  0.02708  0.01896
g_{1} = 3  0.8355  0.07193  0.009358  0.007018  0.003195  0.001651
g_{1} = 4  0.8393  0.04160  0.007018  0.9204 · 10^{−3} 0.6185 · 10^{−3} 0.2454 · 10^{−3}
g_{1} = 5  0.8426  0.02708  0.003195  0.6185 · 10^{−3} 0.7978 · 10^{−4} 0.5011 · 10^{−4}
g_{1} = 6  0.8435  0.01896  0.001651  0.2454 · 10^{−3} 0.5011 · 10^{−4} 0.6383 · 10^{−5}
g_{1} = 7  0.8430  0.01398  0.9350 · 10^{−3} 0.1100 · 10^{−3} 0.1822 · 10^{−4} 0.3840 · 10^{−5}
g_{1} = 8  0.8418  0.01072  0.5675 · 10^{−3} 0.5415 · 10^{−4} 0.7420 · 10^{−5} 0.1315 · 10^{−5}
g_{1} = 9  0.8406  0.008475  0.3638 · 10^{−3} 0.2872 · 10^{−4} 0.3309 · 10^{−5} 0.5001 · 10^{−6}
g_{1} = 10 0.8393  0.006864  0.2438 · 10^{−3} 0.1618 · 10^{−4} 0.1590 · 10^{−5} 0.2072 · 10^{−6}
g_{1} = 11 0.8382  0.005670  0.1694 · 10^{−3} 0.9585 · 10^{−5} 0.8122 · 10^{−6} 0.9226 · 10^{−7}
g_{1} = 12 0.8372  0.004762  0.1213 · 10^{−3} 0.5921 · 10^{−5} 0.4372 · 10^{−6} 0.4366 · 10^{−7}
g_{1} = 13 0.8363  0.004056  0.8909 · 10^{−4} 0.3790 · 10^{−5} 0.2461 · 10^{−6} 0.2177 · 10^{−7}
g_{1} = 14 0.8354  0.003495  0.6691 · 10^{−4} 0.2503 · 10^{−5} 0.1440 · 10^{−6} 0.1136 · 10^{−7}
g_{1} = 15 0.8347  0.003043  0.5122 · 10^{−4} 0.1697 · 10^{−5} 0.8712 · 10^{−7} 0.6173 · 10^{−8}
g_{1} = 16 0.8340  0.002673  0.3986 · 10^{−4} 0.1179 · 10^{−5} 0.5431 · 10^{−7} 0.3474 · 10^{−8}
g_{1} = 17 0.8334  0.002367  0.3149 · 10^{−4} 0.8358 · 10^{−6} 0.3476 · 10^{−7} 0.2017 · 10^{−8}
g_{1} = 18 0.8329  0.002110  0.2520 · 10^{−4} 0.6038 · 10^{−6} 0.2278 · 10^{−7} 0.1204 · 10^{−8}
g_{1} = 19 0.8324  0.001893  0.2041 · 10^{−4} 0.4436 · 10^{−6} 0.1525 · 10^{−7} 0.7378 · 10^{−9}
g_{1} = 20 0.8319  0.001708  0.1670 · 10^{−4} 0.3308 · 10^{−6} 0.1041 · 10^{−7} 0.4624 · 10^{−9}
g_{1} = 21 0.8315  0.001549  0.1380 · 10^{−4} 0.2501 · 10^{−6} 0.7226 · 10^{−8} 0.2960 · 10^{−9}
g_{1} = 22 0.8312  0.001411  0.1150 · 10^{−4} 0.1915 · 10^{−6} 0.5098 · 10^{−8} 0.1931 · 10^{−9}
g_{1} = 23 0.8308  0.001290  0.9663 · 10^{−5} 0.1482 · 10^{−6} 0.3650 · 10^{−8} 0.1282 · 10^{−9}
g_{1} = 24 0.8305  0.001185  0.8177 · 10^{−5} 0.1160 · 10^{−6} 0.2648 · 10^{−8} 0.8649 · 10^{−10}
g_{1} = 25 0.8302  0.001091  0.6966 · 10^{−5} 0.9163 · 10^{−7} 0.1946 · 10^{−8} 0.5923 · 10^{−10}
g_{1} = 26 0.8299  0.001009  0.5971 · 10^{−5} 0.7304 · 10^{−7} 0.1446 · 10^{−8} 0.4113 · 10^{−10}
g_{1} = 27 0.8297 0.9353 · 10^{−3} 0.5147 · 10^{−5} 0.5870 · 10^{−7} 0.1086 · 10^{−8} 0.2893 · 10^{−10}
g_{1} = 28 0.8294 0.8695 · 10^{−3} 0.4461 · 10^{−5} 0.4754 · 10^{−7} 0.8236 · 10^{−9} 0.2060 · 10^{−10}
g_{1} = 29 0.8292 0.8104 · 10^{−3} 0.3885 · 10^{−5} 0.3878 · 10^{−7} 0.6305 · 10^{−9} 0.1483 · 10^{−10}
g_{1} = 30 0.8290 0.7571 · 10^{−3} 0.3400 · 10^{−5} 0.3184 · 10^{−7} 0.4869 · 10^{−9} 0.1079 · 10^{−10}

Table 9. Connected component H^{hyp}(g − 1, g − 1); approximate values of the constants for saddle connections of multiplicity two joining distinct zeroes; g_{1} + g_{2} = g

H_{1}^{hyp}((\overline{g_{1} − 1}) + (\overline{g_{1} − 1})) ≻ H_{1}^{hyp}(\overline{(g_{2} − 1)} + \overline{(g_{2} − 1)}) ≻

g_{2} = 1 g_{2} = 2 g_{2} = 3 g_{2} = 4 g_{2} = 5 g_{2} = 6 g_{2} = 7 g_{2} = 8 g_{2} = 9 g_{2} = 10
g_{1} = 1  0.6250 1.9687 2.9297 4.1122 5.5112 7.1245 8.9513 10.991 13.244 15.709
g_{1} = 2  1.9687 1.1074 2.7191 3.3679 4.1334 5.0053 5.9792 7.0526 8.2244 9.4935
g_{1} = 3  2.9297 2.7191 1.4729 3.3412 3.8411 4.4225 5.0749 5.7932 6.5745 7.4168
g_{1} = 4  4.1122 3.3679 3.3412 1.7750 3.8803 4.2917 4.7662 5.2948 5.8725 6.4961
g_{1} = 5  5.5112 4.1334 3.8411 3.8803 2.0372 4.3611 4.7134 5.1173 5.5653 6.0526
g_{1} = 6  7.1245 5.0053 4.4225 4.2917 4.3611 2.2714 4.7983 5.1083 5.4619 5.8527
g_{1} = 7  8.9513 5.9792 5.0749 4.7662 4.7134 4.7983 2.4849 5.2016 5.4798 5.7955
g_{1} = 8  10.991 7.0526 5.7932 5.2948 5.1173 5.1083 5.2016 2.6821 5.5776 5.8309
g_{1} = 9  13.244 8.2244 6.5745 5.8725 5.5653 5.4619 5.4798 5.5776 2.8663 5.9309
g_{1} = 10 15.709 9.4935 7.4168 6.4961 6.0526 5.8527 5.7955 5.8309 5.9309 3.0396
g_{1} = 11 18.387 10.860 8.3192 7.1637 6.5763 6.2764 6.1434 6.1170 6.1642 6.2652
g_{1} = 12 21.277 12.322 9.2807 7.8740 7.1344 6.7305 6.5198 6.4315 6.4266 6.4821
g_{1} = 13 24.379 13.881 10.301 8.6259 7.7255 7.2128 6.9221 6.7710 6.7143 6.7250
g_{1} = 14 27.694 15.536 11.379 9.4190 8.3487 7.7223 7.3487 7.1334 7.0243 6.9906
g_{1} = 15 31.221 17.288 12.516 10.253 9.0032 8.2577 7.7983 7.5170 7.3547 7.2764
g_{1} = 16 34.961 19.135 13.710 11.127 9.6885 8.8184 8.2697 7.9205 7.7040 7.5806
g_{1} = 17 38.913 21.078 14.962 12.041 10.404 9.4038 8.7624 8.3431 8.0710 7.9019
g_{1} = 18 43.077 23.116 16.271 12.994 11.150 10.014 9.2757 8.7839 8.4548 8.2390
g_{1} = 19 47.453 25.251 17.638 13.987 11.926 10.647 9.8091 9.2425 8.8547 8.5913
g_{1} = 20 52.042 27.481 19.062 15.020 12.731 11.304 10.362 9.7183 9.2702 8.9580
g_{1} = 21 56.842 29.807 20.543 16.092 13.566 11.985 10.935 10.211 9.7007 9.3385
g_{1} = 22 61.856 32.229 22.081 17.203 14.430 12.689 11.527 10.720 10.146 9.7325
g_{1} = 23 67.081 34.746 23.677 18.354 15.323 13.416 12.138 11.246 10.606 10.140
g_{1} = 24 72.518 37.359 25.329 19.544 16.245 14.166 12.768 11.788 11.079 10.559
g_{1} = 25 78.168 40.067 27.039 20.772 17.197 14.939 13.417 12.345 11.567 10.992
g_{1} = 26 84.030 42.871 28.806 22.040 18.177 15.735 14.085 12.919 12.069 11.436
g_{1} = 27 90.104 45.770 30.630 23.347 19.186 16.553 14.771 13.508 12.584 11.893
g_{1} = 28 96.391 48.766 32.510 24.692 20.224 17.394 15.475 14.113 13.113 12.361
g_{1} = 29 102.89 51.856 34.448 26.077 21.291 18.257 16.198 14.733 13.655 12.842
g_{1} = 30 109.60 55.042 36.443 27.500 22.387 19.143 16.939 15.369 14.210 13.334

Appendix B. Saddle Connections Joining a Zero to Itself

B.1. Connected Strata in Genus 4. In the tables below we consider all connected strata in genus g = 4. For each stratum we present the complete list of admissible constructions (= all possible topological configurations of homologous closed geodesics). To make calculations traceable we present for every configuration the orders of possible symmetries Γ and Γ_{−}, the combinatorial constant M, and the result: the constant in quadratic asymptotics for the number of saddle connections (closed geodesics) of this type.

Stratum H(5, 1).

Configurations of closed geodesics.

Degeneration pattern  |Γ_{−}| |Γ| M  c  c approx.
⇒ (\overline{0}, \overline{4}) ⇒  1  1  5 \frac{38125}{15552} 2.45145
→ (\overline{0}, \overline{3}; 1) →  1  1  4  \frac{2240}{243}  9.21811
⇒ (\overline{0 + 3}; 1) ⇒  1  1  4  \frac{320}{243}  1.31687
⇒ (\overline{1 + 2}; 1) ⇒  1  1  4  \frac{320}{243}  1.31687
→ (\overline{1}, \overline{2}; 1) →  1  1  6  \frac{175}{18}  9.72222
→ (\overline{0 + 0}) ⇒ (\overline{0}, \overline{2}) →  1  1  3  \frac{35}{288}  0.121528
→ (\overline{0}, \overline{0}) ⇒ (\overline{0 + 2}) →  1  1  3  \frac{35}{576} 0.0607639
→ (\overline{0}, \overline{0}) ⇒ (\overline{1 + 1}) →  1  1  3  \frac{35}{576} 0.0607639
→ (\overline{0}, \overline{0}) ⇒ (\overline{2 + 0}) →  1  1  3  \frac{35}{576} 0.0607639
→ (\overline{0 + 0}) → (\overline{0}, \overline{1}; 1) →  1  1  2  \frac{175}{486}  0.360082
→ (\overline{0}, \overline{0}) → (\overline{0 + 1}; 1) →  1  1  2  \frac{35}{243}  0.144033
→ (\overline{0 + 0}) ⇒ (\overline{0 + 1}; 1) →  1  1  2  \frac{35}{486} 0.0720165
→ (\overline{0 + 0}) ⇒ (\overline{1 + 0}; 1) →  1  1  2  \frac{35}{486} 0.0720165
→ (\overline{0 + 0}) → (\overline{0 + 0}) ⇒ (\overline{0}, \overline{0}) →  1  1  1  \frac{175}{7776} 0.0225051

Stratum H(4, 1, 1).

Configurations of closed geodesics.

Degeneration pattern  |Γ_{−}| |Γ| M  c  c approx.
⇒ (\overline{0}, \overline{0}; 4) ⇒  2  1  1  \frac{61}{88}  0.693182
⇒ (\overline{0}, \overline{3}; 1) ⇒  1  1  8 \frac{1024}{275}  3.72364
→ (\overline{0}, \overline{2}; 1, 1) →  1  1  3  \frac{432}{55}  7.85455
⇒ (\overline{0 + 2}; 1, 1) ⇒  1  1  3  \frac{54}{55}  0.981818
⇒ (\overline{1 + 1}; 1, 1) ⇒  2  1  \frac{3}{2}  \frac{27}{55}  0.490909
→ (\overline{1}, \overline{1}; 1, 1) →  2  1  2  \frac{224}{55}  4.07273
→ (\overline{0}, \overline{0}) ⇒ (\overline{0}, \overline{2}) →  1  1  6  \frac{27}{275} 0.0981818
⇒ (\overline{0}, \overline{0}) ⇒ (\overline{0 + 2}) ⇒  1  1  6 \frac{27}{1100} 0.0245455
⇒ (\overline{0}, \overline{0}) ⇒ (\overline{1 + 1}) ⇒  2  1  3 \frac{27}{2200} 0.0122727
→ (\overline{0 + 0}) → (\overline{0}, \overline{0}; 1, 1) →  2  1  \frac{1}{2}  \frac{6}{55}  0.109091
→ (\overline{0}, \overline{0}) → (\overline{0 + 0}; 1, 1) →  2  1  \frac{1}{2}  \frac{2}{55}  0.0363636
→ (\overline{0 + 0}) ⇒ (\overline{0 + 0}; 1, 1) →  1  1  1  \frac{2}{55}  0.0363636
→ (\overline{0 + 0}) ⇒ (\overline{0}, \overline{1}; 1) →  1  1  4  \frac{8}{55}  0.145455
→ (\overline{0}, \overline{0}) ⇒ (\overline{0 + 1}; 1) →  1  1  4  \frac{16}{275} 0.0581818
→ (\overline{0}, \overline{0}) ⇒ (\overline{1 + 0}; 1) →  1  1  4  \frac{16}{275} 0.0581818
→ (\overline{0 + 0}) → (\overline{0}, \overline{0}) ⇒ (\overline{0}, \overline{0}) →  2  1  1  \frac{1}{110} 0.00909091
→ (\overline{0 + 0}) ⇒ (\overline{0}, \overline{0}) ⇒ (\overline{0 + 0}) →  2  1  1  \frac{1}{220} 0.00454545

Stratum H(3, 2, 1).

Configurations of closed geodesics.

Degeneration pattern  |Γ_{−}| |Γ| M  c  c approx.
⇒ (\overline{0}, \overline{2}; 2) ⇒  1  1  3  \frac{765}{512}  1.49414
→ (\overline{0}, \overline{0}; 3, 1) →  2  1  \frac{1}{2}  \frac{20}{9}  2.22222
⇒ (\overline{0 + 0}; 3, 1) ⇒  2  1  \frac{1}{2}  \frac{5}{18}  0.277778
⇒ (\overline{0}, \overline{1}; 3) ⇒  1  1  2  \frac{10}{9}  1.11111
→ (\overline{0}, \overline{1}; 2, 1) →  1  1  2  \frac{25}{4}  6.25
⇒ (\overline{0 + 1}; 2, 1) ⇒  1  1  2  \frac{25}{32}  0.78125
⇒ (\overline{1}, \overline{2}; 1) ⇒  1  1  6  \frac{75}{32}  2.34375
→ (\overline{0 + 0}) ⇒ (\overline{0}, \overline{0}; 2) →  1  1  1  \frac{25}{512} 0.0488281
→ (\overline{0}, \overline{0}) ⇒ (\overline{0 + 0}; 2) →  1  1  1  \frac{5}{256} 0.0195313
→ (\overline{0}, \overline{0}) ⇒ (\overline{2}, \overline{0}) →  1  1  3  \frac{15}{256} 0.0585938
⇒ (\overline{0 + 0}) ⇒ (\overline{0}, \overline{2}) ⇒  1  1  3  \frac{15}{512} 0.0292969
→ (\overline{0}, \overline{0}) → (\overline{0}, \overline{1}; 1) →  1  1  2  \frac{25}{144}  0.173611
→ (\overline{0 + 0}) ⇒ (\overline{1}, \overline{0}; 1) →  1  1  2  \frac{25}{288} 0.0868056
→ (\overline{0}, \overline{0}) ⇒ (\overline{1}, \overline{1}) →  1  1  4  \frac{5}{72}  0.0694444
⇒ (\overline{0 + 0}) ⇒ (\overline{0 + 1}; 1) ⇒  1  1  2  \frac{5}{288} 0.0173611
→ (\overline{0}, \overline{0}) → (\overline{0 + 0}) ⇒ (\overline{0}, \overline{0}) →  1  1  1 \frac{25}{2304} 0.0108507
→ (\overline{0 + 0}) ⇒ (\overline{0 + 0}) ⇒ (\overline{0}, \overline{0}) →  1  1  1 \frac{25}{4608} 0.00542535

Stratum H(3, 1, 1, 1).

Configurations of closed geodesics.

Degeneration pattern  |Γ_{−}| |Γ| M  c  c approx.
⇒ (\overline{0}, \overline{0}; 3, 1) ⇒  2  1  3  \frac{60}{31}  1.93548
⇒ (\overline{0}, \overline{2}; 1, 1) ⇒  1  1  9 \frac{2025}{496} 4.08266
→ (\overline{0}, \overline{1}; 1, 1, 1) →  1  1  2 \frac{1575}{248} 6.35081
⇒ (\overline{0 + 1}; 1, 1, 1) ⇒  1  1  2  \frac{175}{248} 0.705645
⇒ (\overline{0}, \overline{0}) ⇒ (\overline{0}, \overline{2}) ⇒  1  1 18 \frac{405}{7936} 0.0510333
→ (\overline{0 + 0}) ⇒ (\overline{0}, \overline{0}; 1, 1) →  1  1  3 \frac{225}{1984} 0.113407
→ (\overline{0}, \overline{0}) ⇒ (\overline{0 + 0}; 1, 1) →  1  1  3 \frac{75}{1984} 0.0378024
→ (\overline{0}, \overline{0}) ⇒ (\overline{0}, \overline{1}; 1) →  1  1 12 \frac{75}{496}  0.15121
⇒ (\overline{0}, \overline{0}) ⇒ (\overline{0 + 1}; 1) ⇒  1  1 12 \frac{15}{496} 0.0302419
→ (\overline{0 + 0}) ⇒ (\overline{0}, \overline{0}) ⇒ (\overline{0}, \overline{0}) →  1  1  6 \frac{75}{7936} 0.0094506

Stratum H(2, 2, 1, 1).

Configurations of closed geodesics.

Degeneration pattern  |Γ_{−}| |Γ| M  c  c approx.
⇒ (\overline{0}, \overline{0}; 2, 2) ⇒  2  1  1  \frac{85}{131}  0.648855
→ (\overline{0}, \overline{0}; 2, 1, 1) →  2  1  1  \frac{600}{131}  4.58015
⇒ (\overline{0 + 0}; 2, 1, 1) ⇒  2  1  1  \frac{200}{393}  0.508906
⇒ (\overline{0}, \overline{1}; 2, 1) ⇒  1  1  8  \frac{1600}{393}  4.07125
⇒ (\overline{1}, \overline{1}; 1, 1) ⇒  2  1  4  \frac{5600}{3537}  1.58326
→ (\overline{0}, \overline{0}) ⇒ (\overline{0}, \overline{0}; 2) →  1  1  4  \frac{25}{393}  0.0636132
⇒ (\overline{0 + 0}) ⇒ (\overline{0}, \overline{0}; 2) ⇒  2  1  2  \frac{25}{1572} 0.0159033
⇒ (\overline{0}, \overline{0}) ⇒ (\overline{0 + 0}; 2) ⇒  2  1  2  \frac{5}{786} 0.00636132
→ (\overline{0}, \overline{0}) → (\overline{0}, \overline{0}; 1, 1) →  2  1  1  \frac{100}{1179} 0.0848176
→ (\overline{0}, \overline{0}) ⇒ (\overline{1}, \overline{0}; 1) →  1  1  8  \frac{400}{3537}  0.11309
⇒ (\overline{0 + 0}) ⇒ (\overline{0 + 0}; 1, 1) ⇒  2  1  1  \frac{25}{3537} 0.00706814
⇒ (\overline{0 + 0}) ⇒ (\overline{0}, \overline{1}; 1) ⇒  1  1  8  \frac{200}{3537} 0.0565451
⇒ (\overline{0}, \overline{0}) ⇒ (\overline{1}, \overline{1}) ⇒  2  1  8  \frac{80}{3537}  0.022618
→ (\overline{0}, \overline{0}) → (\overline{0}, \overline{0}) ⇒ (\overline{0}, \overline{0}) →  2  1  2  \frac{25}{3537} 0.00706814
→ (\overline{0}, \overline{0}) ⇒ (\overline{0 + 0}) ⇒ (\overline{0}, \overline{0}) →  2  1  2  \frac{25}{7074} 0.00353407
⇒ (\overline{0 + 0}) ⇒ (\overline{0 + 0}) ⇒ (\overline{0}, \overline{0}) ⇒  2  1  2 \frac{25}{14148} 0.00176703

Stratum H(2, 1, 1, 1, 1).

Configurations of closed geodesics.

Degeneration pattern  |Γ_{−}| |Γ| M  c  c approx.
⇒ (\overline{0}, \overline{0}; 2, 1, 1) ⇒  2  1  6  \frac{18}{5}  3.6
→ (\overline{0}, \overline{0}; 1, 1, 1, 1) →  2  1  \frac{1}{2}  \frac{7}{3}  2.33333
⇒ (\overline{0 + 0}; 1, 1, 1, 1) ⇒  2  1  \frac{1}{2}  \frac{7}{30}  0.233333
⇒ (\overline{0}, \overline{1}; 1, 1, 1) ⇒  1  1  8  \frac{56}{15}  3.73333
⇒ (\overline{0}, \overline{0}) ⇒ (\overline{0}, \overline{0}; 2) ⇒  2  1 12 \frac{1}{40}  0.025
→ (\overline{0}, \overline{0}) ⇒ (\overline{0}, \overline{0}; 1, 1) →  1  1 12 \frac{2}{15}  0.133333
⇒ (\overline{0 + 0}) ⇒ (\overline{0}, \overline{0}; 1, 1) ⇒  2  1  6  \frac{1}{30} 0.0333333
⇒ (\overline{0}, \overline{0}) ⇒ (\overline{0 + 0}; 1, 1) ⇒  2  1  6  \frac{1}{90} 0.0111111
⇒ (\overline{0}, \overline{0}) ⇒ (\overline{0}, \overline{1}; 1) ⇒  1  1 48 \frac{4}{45} 0.0888889
→ (\overline{0}, \overline{0}) ⇒ (\overline{0}, \overline{0}) ⇒ (\overline{0}, \overline{0}) →  2  1 12 \frac{1}{180} 0.00555556
⇒ (\overline{0 + 0}) ⇒ (\overline{0}, \overline{0}) ⇒ (\overline{0}, \overline{0}) ⇒  2  1 12 \frac{1}{360} 0.00277778

Stratum H(1, 1, 1, 1, 1, 1).

Configurations of closed geodesics.

Degeneration pattern  |Γ_{−}| |Γ| M  c  c approx.
⇒ (\overline{0}, \overline{0}; 1, 1, 1, 1) ⇒  2  1  15 \frac{3150}{377} 8.35544
⇒ (\overline{0}, \overline{0}) ⇒ (\overline{0}, \overline{0}; 1, 1) ⇒  2  1 180 \frac{90}{377} 0.238727
⇒ (\overline{0}, \overline{0}) ⇒ (\overline{0}, \overline{0}) ⇒ (\overline{0}, \overline{0}) ⇒  2  3 120 \frac{5}{754} 0.0066313

MODULI OF ABELIAN DIFFERENTIALS: THE PRINCIPAL BOUNDARY 85

B.2. Closed Geodesics for Nonconnected Strata in Genus 4

Component H^{hyp}(6).

Degeneration pattern |Γ_{−}| |Γ| c · ζ(2) c · ζ(2) approx.

→ (\overline{2}, \overline{2}) →  2  1  \frac{384}{25}  15.36
⇒ (\overline{2 + 2}) ⇒  2  1  \frac{6}{1}  6.
→ (\overline{0}, \overline{0}) → (\overline{1 + 1}) →  2  1  \frac{84}{25}  3.36
→ (\overline{0 + 0}) → (\overline{1}, \overline{1}) →  2  1  \frac{896}{225}  3.98222
Component H^{even}(6).
→ (\overline{0}, \overline{4}) →  1  1  \frac{30375}{4096}  7.41577
⇒ (\overline{0 + 4}) ⇒  1  1  \frac{10125}{8192}  1.23596
⇒ (\overline{1 + 3}) ⇒  1  1  \frac{875}{256}  3.41797
→ (\overline{2}, \overline{2}) →  2  1  \frac{405}{128}  3.16406
→ (\overline{1}, \overline{3}) →  1  1  \frac{15}{1}  15.
→ (\overline{0 + 0}) → (\overline{0}, \overline{2}) →  1  1  \frac{2835}{2048}  1.38428
→ (\overline{0}, \overline{0}) → (\overline{0 + 2}) →  1  1  \frac{2835}{4096}  0.692139
→ (\overline{0 + 0}) ⇒ (\overline{1 + 1}) →  1  1  \frac{2835}{8192}  0.346069
Component H^{odd}(6).
→ (\overline{0}, \overline{4}) →  1  1  \frac{350}{27}  12.963
⇒ (\overline{0 + 4}) ⇒  1  1  \frac{175}{81}  2.16049
⇒ (\overline{1 + 3}) ⇒  1  1  \frac{25}{32}  0.78125
⇒ (\overline{2 + 2}) ⇒  2  1  \frac{175}{162}  1.08025
→ (\overline{2}, \overline{2}) →  2  1  \frac{105}{16}  6.5625
→ (\overline{1}, \overline{3}) →  1  1  \frac{256}{27}  9.48148
→ (\overline{0 + 0}) ⇒ (\overline{0 + 2}) →  1  1  \frac{7}{32}  0.21875
→ (\overline{0 + 0}) ⇒ (\overline{2 + 0}) →  1  1  \frac{7}{32}  0.21875
→ (\overline{0 + 0}) → (\overline{1}, \overline{1}) →  2  1  \frac{7}{27}  0.259259
→ (\overline{0 + 0}) → (\overline{0 + 0}) → (\overline{0}, \overline{0}) →  2  1  \frac{35}{432}  0.0810185
→ (\overline{0 + 0}) → (\overline{0 + 0}) ⇒ (\overline{0 + 0}) →  2  1  \frac{35}{864}  0.0405093

86 ALEX ESKIN, HOWARD MASUR, AND ANTON ZORICH

Configurations of closed geodesics. Component H^{hyp}(3, 3).

Degeneration pattern |Γ_{−}| |Γ| c · ζ(2) c · ζ(2) approx.

⇒ (\overline{2}, \overline{2}) ⇒  2  1  \frac{15}{2}  7.5
→ (\overline{0}, \overline{0}) → (\overline{1}, \overline{1}) →  2  1  \frac{35}{9}  3.88889
Component H^{nonhyp}(3, 3).

Degeneration pattern |Γ_{−}| |Γ| c · ζ(2) c · ζ(2) approx.

⇒ (\overline{2}, \overline{2}) ⇒  2  1  \frac{3699}{1120}  3.30268
→ (\overline{0}, \overline{1}; 3) →  1  1  \frac{64}{5}  12.8
⇒ (\overline{0 + 1}; 3) ⇒  1  1  \frac{64}{35}  1.82857
→ (\overline{0 + 0}) ⇒ (\overline{2}, \overline{0}) →  1  1  \frac{27}{80}  0.3375
→ (\overline{0}, \overline{0}) → (\overline{1}, \overline{1}) →  2  1  \frac{1}{5}  0.2
→ (\overline{0}, \overline{0}) → (\overline{0 + 0}) ⇒ (\overline{0 + 0}) →  2  1  \frac{1}{32}  0.03125
Component H^{even}(4, 2).

Degeneration pattern |Γ_{−}| |Γ| c · ζ(2) c · ζ(2) approx.

→ (\overline{0}, \overline{0}; 4) →  2  1  \frac{4}{3}  1.33333
⇒ (\overline{0 + 0}; 4) ⇒  2  1  \frac{4}{21}  0.190476
→ (\overline{0}, \overline{2}; 2) →  1  1  \frac{256}{45}  5.68889
⇒ (\overline{0 + 2}; 2) ⇒  1  1  \frac{256}{315}  0.812698
⇒ (\overline{1 + 1}; 2) ⇒  2  1  \frac{8}{9}  0.888889
⇒ (\overline{1}, \overline{3}) ⇒  1  1  \frac{32768}{8505}  3.85279
→ (\overline{1}, \overline{1}; 2) →  2  1  \frac{128}{27}  4.74074
→ (\overline{0 + 0}) → (\overline{0}, \overline{0}; 2) →  2  1  \frac{8}{27}  0.296296
→ (\overline{0}, \overline{0}) → (\overline{0 + 0}; 2) →  2  1  \frac{16}{135}  0.118519
→ (\overline{0}, \overline{0}) → (\overline{0}, \overline{2}) →  1  1  \frac{32}{45}  0.711111
⇒ (\overline{0 + 0}) ⇒ (\overline{1 + 1}) ⇒  2  1  \frac{2}{45}  0.0444444
→ (\overline{0 + 0}) ⇒ (\overline{1}, \overline{1}) →  1  1  \frac{256}{1215}  0.2107

Component H^{odd}(4, 2).

Degeneration pattern |Γ_{−}| |Γ| c · ζ(2) c · ζ(2) approx.

→ (\overline{0}, \overline{0}; 4) →  2  1  \frac{49}{18}  2.72222
⇒ (\overline{0 + 0}; 4) ⇒  2  1  \frac{7}{18}  0.388889
→ (\overline{0}, \overline{2}; 2) →  1  1  \frac{147}{16}  9.1875
⇒ (\overline{0 + 2}; 2) ⇒  1  1  \frac{21}{16}  1.3125
⇒ (\overline{1 + 1}; 2) ⇒  2  1  \frac{3}{10}  0.3
⇒ (\overline{1}, \overline{3}) ⇒  1  1  \frac{128}{45}  2.84444
→ (\overline{1}, \overline{1}; 2) →  2  1  \frac{7}{2}  3.5
→ (\overline{0 + 0}) ⇒ (\overline{0 + 0}; 2) →  1  1  \frac{7}{80}  0.0875
⇒ (\overline{0 + 0}) ⇒ (\overline{0 + 2}) ⇒  1  1  \frac{21}{320}  0.065625
→ (\overline{0 + 0}) ⇒ (\overline{1}, \overline{1}) →  1  1  \frac{7}{45}  0.155556
→ (\overline{0 + 0}) → (\overline{0}, \overline{0}) → (\overline{0}, \overline{0}) →  2  1  \frac{7}{144}  0.0486111
→ (\overline{0 + 0}) ⇒ (\overline{0 + 0}) ⇒ (\overline{0 + 0}) →  2  1  \frac{7}{576}  0.0121528

Component H^{even}(2, 2, 2).
→ (\overline{0}, \overline{0}; 2, 2) →  2  1  \frac{180}{37}  4.86486
⇒ (\overline{0 + 0}; 2, 2) ⇒  2  1  \frac{45}{74}  0.608108
⇒ (\overline{1}, \overline{1}; 2) ⇒  2  1  \frac{225}{37}  6.08108
→ (\overline{0}, \overline{0}) → (\overline{0}, \overline{0}; 2) →  2  1  \frac{225}{296}  0.760135
⇒ (\overline{0 + 0}) ⇒ (\overline{1}, \overline{1}) ⇒  2  1  \frac{5}{37}  0.135135

Component H^{odd}(2, 2, 2).
→ (\overline{0}, \overline{0}; 2, 2) →  2  1  \frac{252}{31}  8.12903
⇒ (\overline{0 + 0}; 2, 2) ⇒  2  1  \frac{63}{62}  1.01613
⇒ (\overline{1}, \overline{1}; 2) ⇒  2  1  \frac{144}{31}  4.64516
⇒ (\overline{0 + 0}) ⇒ (\overline{0 + 0}; 2) ⇒  2  1  \frac{9}{155}  0.0580645
⇒ (\overline{0 + 0}) ⇒ (\overline{1}, \overline{1}) ⇒  2  1  \frac{16}{155}  0.103226
→ (\overline{0}, \overline{0}) → (\overline{0}, \overline{0}) → (\overline{0}, \overline{0}) →  2  3  \frac{4}{93}  0.0430108
⇒ (\overline{0 + 0}) ⇒ (\overline{0 + 0}) ⇒ (\overline{0 + 0}) ⇒  2  3  \frac{1}{186}  0.00537634

References

[At] M. Atiyah, Riemann surfaces and spin structures, Ann. scient. ÉNS 4^{e} série 4 (1971), 47–62.

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