The geometry of surface-by-free groups

Benson Farb^{∗} and Lee Mosher^{†}

August 29, 2000

Abstract We show that every word hyperbolic, surface-by-(noncyclic) free group Γ is as rigid as possible: the quasi-isometry group of Γ equals the abstract commensurator group Comm(Γ), which in turn contains Γ as a finite-index subgroup. As a corollary, two such groups are quasi-isometric if and only if they are commensurable, and any finitely-generated group quasi-isometric to Γ must be weakly commensurable with Γ. We use quasi-isometries to compute Comm(Γ) explicitly, an example of how quasi-isometries can actually detect finite-index information. The proofs of these theorems involve ideas from coarse topology, Teichmüller geometry, pseudo-Anosov dynamics, and singular solv geometry.

1 Introduction

Let Σ_{g} be a closed surface of genus g ≥ 2, and let M(Σ_{g}) = π_{0}(Homeo(Σ_{g})) denote the mapping class group of Σ_{g}. A Schottky subgroup H of M(Σ_{g}) is a free group of pseudo-Anosov mapping classes whose action on the Teichmüller space T (Σ_{g}) is “weak convex cocompact”—the group H has a limit set in Thurston’s boundary of T (Σ_{g}), when rank(H) ≥ 2 this limit set is a Cantor set, and the action of H on the “weak convex hull” of the limit set is cocompact. Schottky subgroups of M(Σ_{g}) exist in abundance: given any collection {ϕ_{1}, …, ϕ_{r}} of pairwise independent pseudo-Anosov mapping classes of Σ_{g}, for any sufficiently large positive integers a_{1}, …, a_{r} the elements {ϕ_{1}^{a_{1}}, …, ϕ_{r}^{a_{r}}} freely generate a Schottky subgroup. See §3 for a review of Schottky groups, taken from [Mos97] and [FM00b].

Given g ≥ 2 and a Schottky subgroup H < M(Σ_{g}) ≈ Out(π_{1}(Σ_{g})), one can construct a group Γ_{H} given by the split extension

1 → π_{1}(Σ_{g}) → Γ_{H} → H → 1

∗Supported in part by NSF grant DMS 9704640 and by a Sloan Foundation Fellowship.

†Supported in part by NSF grant DMS 9504946.

The groups Γ_{H} are precisely the surface-by-free groups which are word hyperbolic [FM00b], and the construction of [Mos97] shows that they are abundant.

We are interested in studying quasi-isometries of the groups Γ_{H} for several reasons: the Γ_{H} provide basic examples of rigidity theorems for word hyperbolic groups outside the context of negatively curved manifolds (see also [Bou00], [KK98]); they are examples of groups which can be viewed as phase spaces of dynamical systems arising from hyperbolic endomorphisms of manifolds (see also [FM98], [FM99], [FM00a]); and they provide examples of groups which are as rigid as possible in a very concrete sense (see Theorem 1.3 below). In particular, each Γ_{H} has finite index in its own abstract commensurator Comm(Γ_{H}) as well as in its quasi-isometry group. The computation of Comm(Γ_{H}) given in Theorem 1.3 is explicit, one of the few instances outside lattices in Lie groups where this has been done.

Finally, we propose the general problem of studying the asymptotic geometry of extensions of surface groups Σ. These groups exhibit a beautiful and rich geometry which is encoded by a subgroup of M(Σ) acting on the Teichmüller space T (Σ). Some elements of the geometry of such groups can be found in [Mos96], [Mit98] and [FM00b], but our needs will require a somewhat involved account of this theory for extensions of surface groups by Schottky groups, which we give in §4.

Statement of results

Our first theorem gives a complete classification of the groups Γ_{H} up to quasiisometry. Although the usual quasi-isometry invariants such as growth, ends, isoperimetric functions, etc., are the same for all of these groups, they are only quasi-isometric under the strictest algebraic conditions.

Given a Schottky subgroup H of M(Σ), we will show in §7.2 that there is a smallest 2-orbifold covered by Σ, denoted O_{H}, such that H descends via the covering map Σ → O_{H} to a Schottky subgroup of M(O_{H}), still denoted H. The orbifold O_{H} plays an important role in describing the quasi-isometry class of the surface-by-free group Γ_{H}.

Theorem 1.1 (Classification Theorem). Given g_{1}, g_{2} ≥ 2, let H_{1} < M(Σ_{g_{1}}) and H_{2} < M(Σ_{g_{2}}) by Schottky subgroups of rank ≥ 2. The following are equivalent:

(1) The surface-by-free groups Γ_{H_{1}} and Γ_{H_{2}} are quasi-isometric.

(2) Γ_{H_{1}} and Γ_{H_{2}} are abstractly commensurable, meaning that they have finiteindex subgroups which are isomorphic.

(3) There is an isomorphism O_{H_{1}} ≈ O_{H_{2}} such that in the group M(O_{H_{1}}) = M(O_{H_{2}}) the Schottky subgroups H_{1} and H_{2} are commensurable, meaning that H_{1} ∩ H_{2} has finite index in each of H_{1} and H_{2}.

(4) There is an isomorphism O_{H_{1}} ≈ O_{H_{2}} such that in the group M(O_{H_{1}}) = M(O_{H_{2}}) the Schottky subgroups H_{1} and H_{2} have the same limit set in the Thurston boundary of the Teichmüller space T (O_{H_{1}}) = T (O_{H_{2}}).

The next theorem shows that each of the groups Γ_{H} is determined among all finitely-generated groups by its asymptotic geometry, up to finite data.

Theorem 1.2 (Quasi-isometric rigidity). Let Γ_{H} be a surface-by-free group with H < M(Σ_{g}) a Schottky group of rank ≥ 2. If G is any finitely-generated group which is quasi-isometric to Γ_{H}, then there is a finite normal subgroup F < G such that G/F is abstractly commensurable to a surface-by-free group. Combining with Theorem 1.1 it follows that G/F is abstractly commensurable to Γ_{H}.

Remark. We emphasize that it is essential for our methods that rank(H) ≥ 2 in the statements of Theorems 1.1 and 1.2. Indeed, when H is a rank 1 Schottky subgroup, in other words an infinite cyclic subgroup generated by a pseudo-Anosov mapping class, then Γ_{H} is the fundamental group of a closed hyperbolic 3-manifold that fibers over the circle [Ota96], and hence all the groups Γ_{H} with H Schottky of rank 1 are quasi-isometric to each other and to H^{3}. Moreover, the restatement of Theorem 1.2 (minus the last sentence) for H of rank 1 is equivalent to Thurston’s virtual surface bundle conjecture for closed hyperbolic 3-manifolds. ^{1}

Theorem 1.2 and (most of) Theorem 1.1 follow from our main result, Theorem 1.3 below.

Commensurations and quasi-isometries. Recall that a commensuration of Γ is an isomorphism between finite index subgroups of Γ. Composition of two commensurations is defined on a further finite index subgroup. Two commensurations are equivalent if they agree on a common finite index subgroup. Composition of equivalence classes gives a well-defined group operation, and we thereby obtain the abstract commensurator group Comm(Γ) of the group Γ.

The quasi-isometry group QI(Γ) is the group of coarse equivalence classes of self quasi-isometries of Γ (endowed with any word metric), where two quasi-isometries are coarsely equivalent if they have finite distance in the sup norm.

1This equivalence, however, hides the following fact: Theorem 1.1 provides a commensuration from Γ_{H_{1}} to Γ_{H_{2}} coarsely taking π_{1}(Σ_{g_{2}}) to π_{1}(Σ_{g_{2}}), a situation which is often impossible for rank 1; see §8.2. Hence the proofs of Theorems 1.1 and 1.2 cannot shed light on the virtual surface bundle conjecture.

In general it is difficult to compute the abstract commensurator of a group. The computation for irreducible lattices in semisimple groups G ≠ PSL(2, R) is the content of Mostow Rigidity together with theorems of Borel and Margulis (see e.g. [Zim84]). Our main result, Theorem 1.3, says that the groups Comm(Γ_{H}) and QI(Γ_{H}) are isomorphic, and gives an explicit computation of these groups. This is the first time we know of where quasi-isometries are used to compute the abstract commensurator group. While our expression for Comm(Γ_{H}) is purely algebraic, we do not know how to do the computation algebraically.

Recall that given groups K < Q, the (relative) commensurator of K in Q, denoted Comm_{Q}(K), is the subgroup of all q ∈ Q such that conjugation by q takes some finite index subgroup of K to another, or equivalently K ∩ qKq^{−1} has finite index in both K and qKq^{−1}; this subgroup is also known as the virtual normalizer of K in Q. The group C = Comm_{M(O_{H})}(H), the relative commensurator of H in M(O_{H}), plays a key role in computing the abstract commensurator of Γ_{H}:

Theorem 1.3 (Computation of QI(Γ_{H}) and Comm(Γ_{H})). Given g ≥ 2 and a Schottky group H < M(Σ_{g}) of rank ≥ 2, the natural homomorphism

Comm(Γ_{H}) → QI(Γ_{H})

is an isomorphism. Furthermore, these groups are isomorphic to the group Γ_{C} given explicitly by the short exact sequence

1 → π_{1}(O_{H}) → Γ_{C} → Comm_{M(O_{H})}(H) → 1

Moreover, C = Comm_{M(O_{H})}(H) contains H as a finite index subgroup. In particular, Γ_{H} has finite index in Comm(Γ_{H}) ≈ QI(Γ_{H}).

This theorem is proved in Sections 7.3 and 7.4.

The fact that Γ_{H} has finite index in QI(Γ_{H}) and also in Comm(Γ_{H}) shows that the groups Γ_{H} are extremely rigid. Among irreducible lattices in semisimple Lie groups, this phenomenon holds for the noncocompact, nonarithmetic lattices and for no other lattice; see [Mar91] for the commensurator statement, and [Sch96] and e.g. [Far97] for the quasi-isometry statement.

Outline of the proof of Theorem 1.3

After some preliminary material, we begin in §4 by constructing a quasi-isometric model space X_{H} on which Γ_{H} acts properly cocompactly by isometries. A Schottky subgroup H < M(Σ) gives an H-equivariant embedding of the Cayley graph T_{H} of H, a tree, in the Teichmüller space T (Σ). This embedding gives a Σ-bundle over T_{H}, each fiber carrying a hyperbolic structure representing the point of T_{H} ⊂ T (Σ) over which that fiber lies. The universal cover of this Σ-bundle is our model space X_{H}, an H^{2}-bundle over the tree T_{H}.

An isometry of Γ_{H} acts on X_{H}, permuting or “respecting” various patterns of geometric objects, typically foliations. Indeed, the same patterns respected by isometries of X_{H} are also respected by the finite index supergroup Γ_{C} which is defined by the short exact sequence given in Theorem 1.3; see §7.3 for the precise definition of Γ_{C}.

The rest of the proof is devoted to showing that an arbitrary quasi-isometry f: X_{H} → X_{H} coarsely respects so many patterns that it must be close to an element of Γ_{C}. Using coarse topology, we show in §4 that f permutes the collection of “hyperplanes” P_{w}; these are the H^{2}-bundles over bi-infinite lines w in T_{H}. The Schottky property is used to relate each line w to a Teichmüller geodesic, which in turn allows us to impose extra structure on the hyperplane P_{w}: a “pseudo-Anosov flow” and a singular solv structure. In §5 we use this structure to prove that f coarsely respects several dynamically defined foliations associated to these flows, such as the stable and unstable foliations.

In §6, we apply the above together with R. Schwartz’s geodesic pattern rigidity [Sch97] to show that the quasi-isometry f actually permutes the collection of periodic hyperplanes, i.e. those P_{w} with w a bi-infinite periodic line in the tree T_{H}.

In §7 we study how the quasi-isometry f acts on the limit set of H in Thurston’s boundary of T (Σ), the space of projective measured foliations. This limit set is a Cantor set, and the set of periodic hyperplanes gives a countable dense subset which is preserved by the action of f. This information is then used to show that f is close to an element of Γ_{C}.

The final parts of the proofs of all of the main theorems are contained in §7.

Surfaces versus orbifolds. Our main theorems are stated solely for closed, oriented surfaces Σ_{g} of genus g ≥ 2, and Schottky subgroups H of M(Σ_{g}). But the conclusions of these theorems force us to consider a wider realm: closed 2-orbifolds O and virtual Schottky subgroups of M(O); see e.g. Theorem 1.3 in which, as the proof will show, Comm_{M(O_{H})}(H) is a virtual Schottky subgroup of M(O_{H}). Various results that we will need about (virtual) Schottky subgroups of orbifold mapping class groups are formulated in [FM00b].

This raises the question of whether the quasi-isometry classes of (orbifold)- by-(virtual Schottky) groups constitute a wider universe than the quasi-isometry classes of (surface)-by-(Schottky) groups. The answer is no: these universes are identical. The proof is given in Section 8.1.

Acknowledgements We are grateful to Kevin Whyte for crucial help in proving Theorem 7.4.

2 Preliminaries

In this section we briefly review some facts about Teichmüller space and about quasi-isometries. For details we refer to [Abi80], [FLP^{+}79], [IT92].

For most of the paper, while we concentrate on the proof of Theorem 1.3, we shall fix the genus g and denote Σ = Σ_{g}.

2.1 Teichmüller space and mapping class groups

The Teichmüller space of Σ, denoted T = T (Σ), has two equivalent descriptions, related to each other by Riemann’s uniformization theorem: T is the space of conformal structures on Σ modulo isotopy; or it is the space of hyperbolic structures on Σ modulo isotopy. A topology and a real analytic structure on T is specified by the geodesic length embedding T → [0, ∞)^{C} where C is the set of isotopy classes of nontrivial simple closed curves on Σ, and a hyperbolic structure on Σ determines an element of [0, ∞)^{C} by taking the length of the unique closed geodesic in each isotopy class. With respect to this topology, T is homeomorphic to Euclidean space of dimension 6g − 6.

The mapping class group of Σ, denoted M = M(Σ), is the group π_{0}(Homeo(Σ) = Homeo(Σ)/ Homeo_{0}(Σ) where Homeo(Σ) is the group of all self-homeomorphisms of Σ and Homeo_{0}(Σ) is the normal subgroup of self-homeomorphisms isotopic to the identity.

The group M acts real analytically on T . Also, M acts on C and so on [0, ∞)^{C}, and the embedding T → [0, ∞)^{C} is M-equivariant. The action of M on T is properly discontinuous. The quotient orbifold M = M(Σ) = T /M is called the moduli space of Σ. A subset of T is said to be cobounded if its image under the quotient map T → M is bounded.

2.2 Measured foliations

A measured foliation on Σ is a foliation on the complement of a finite set of singularities, together with a positive, transverse Borel measure, such that each singularity is an n-pronged singularity for some n ≥ 3, locally modelled on the horizontal measured foliation of the quadratic differential z^{n−2} dz in the complex plane. A saddle connection of a measured foliation is leaf segment which connects two distinct singularities, and Whitehead equivalence is the equivalence relation on the set of measured foliations generated by isotopy and the collapse of saddle connections. The measured foliation space of Σ, denoted MF = MF (Σ), is the space of Whitehead equivalence classes of measured foliations on Σ. A topology on MF is specified by the “transverse measure embedding” MF → [0, ∞)^{C}, where a measured foliation determines an element of [0, ∞)^{C} by taking the infimum of the transverse measures of representatives of C.

Given a measured foliation F and r ∈ (0, ∞), multiplying the transverse measure on F by r gives a new measured foliation denoted rF. This gives a free action of (0, ∞) on MF, whose quotient space is the space of projective measured foliations on Σ, denoted PMF = PMF (Σ).

The embedding T ↪ [0, ∞)^{C}, composed with the projectivization map [0, ∞)^{C} → P[0, ∞)^{C}, produces an embedding T ↪ P[0, ∞)^{C}. The embedding MF ↪ [0, ∞)^{C} induces an embedding PMF ↪ P[0, ∞)^{C}. Thurston’s Compactification Theorem [FLP^{+}79] says that image of \overline{T} = T ∪ PMF in P[0, ∞)^{C} is a closed ball of dimension 6g − 6, whose interior is T and whose boundary sphere is PMF.

2.3 Geodesics in T

The Teichmüller metric and its geodesics are usually described in terms of holomorphic quadratic differentials on Riemann surfaces. Using results of Gardiner and Masur [GM91] and of Hubbard and Masur [HM79], the metric can be presented directly in terms of measured foliations.

Consider of pair of measured foliations F_{x}, F_{y} which are transverse, meaning that they have the same singular set, at each singularity F_{x} and F_{y} have the same number of prongs, they are transverse in the usual sense away from the singularities, and near an n-pronged singularity they are locally modelled on the horizontal and vertical measured foliations of the quadratic differential z^{n−2}dz on the complex plane. Let |dy|, |dx| denote the transverse measures on F_{x}, F_{y}, respectively; the leaves of F_{x} should be visualized as horizontal lines with transverse measure |dy|, and the leaves of F_{y} as vertical lines with transverse measure |dx|. The formula dx^{2} + dy^{2} defines a singular Euclidean metric on Σ denoted μ(F_{x}, F_{y}), with total area

Area(μ) = ∫_{Σ} |dx| |dy| < ∞

Underlying the metric μ(F_{x}, F_{y}) is a conformal structure on the complement of the singularities, but the singularities are removable and so we obtain a conformal structure on Σ and a point in T denoted σ(F_{x}, F_{y}). This gives a well-defined map from a certain subset of MF × MF to T . Namely, letting U ⊂ MF × MF denote the set of pairs (ξ, η) which are represented by a transverse pair (F_{x}, F_{y}) of measured foliations, it follows that σ(ξ, η) = σ(F_{x}, F_{y}) is well-defined independent of the choice of the representative pair (F_{x}, F_{y}), defining a map U → T (see [GM91], Theorem 3.1).

A transverse pair F_{x}, F_{y} is normalized if Area(μ(F_{x}, F_{y})) = 1. Let U_{0} ⊂ U be the subset represented by normalized transverse pairs. For each (F_{x}, F_{y}) ∈ U_{0}, the map t ↦ γ(t) = σ(e^{t}F_{x}, e^{−t}F_{y}) is a real analytic embedding of R in T ; the image of this embedding depends only on the projective classes ξ = PF_{x}, η = PF_{y} and is denoted \overleftrightarrow{(ξ, η)} = \overleftrightarrow{(PF_{x}, PF_{y})}. Teichmüller’s Theorem [Abi80] says that any two points p ≠ q ∈ T are contained in a unique such line \overleftrightarrow{(PF_{x}, PF_{y})}; moreover, if s, t ∈ R are such that p = σ(e^{t}F_{x}, e^{−t}F_{y}) and q = σ(e^{s}F_{x}, e^{−s}F_{y}), then the formula

d(p, q) = |s − t|

gives a well-defined metric on T , known as the Teichmüller metric. Each line \overleftrightarrow{(ξ, η)} then becomes a bi-infinite geodesic in T . If we restrict the parameterization γ(t) = σ(e^{t}F_{x}, e^{−t}F_{y}) to the half-line t ∈ [0, ∞) then we obtain a geodesic ray in T and we call the point PF_{x} ∈ PMF the ending foliation of the ray; if σ = γ(0) and ξ = PF_{x} then this ray is denoted \overrightarrow{[σ, ξ)}. For any Teichmüller line \overleftrightarrow{(ξ, η)} the two points ξ, η are called the ending foliations of the line. Any two points σ, τ ∈ T are the endpoints of a unique finite geodesic segment, denoted \overline{στ}.

With respect to the Teichmüller metric, T is a complete metric space on which M acts by isometries. Royden’s Theorem [Roy71] says, when the surface Σ is closed and oriented, that the homomorphism M → Isom(T ) is an isomorphism, except for a small kernel on certain small surfaces: on Σ_{2} the single nontrivial element of the kernel being the hyperelliptic involution of Σ_{2}.

Given σ ∈ T and ξ ∈ PMF there is exactly one ray in T with endpoint σ and with ending foliation ξ; we denote this ray \overrightarrow{[σ, ξ)}. This gives a one-to-one correspondence between T × PMF and geodesic rays in T . Given ξ, η ∈ PMF, there exists at most one geodesic in T with ending foliations ξ, η, and it exists if and only if (ξ, η) is in the image of U_{0} ⊂ MF × MF under the projectivization map MF × MF → PMF × PMF. If this geodesic exists we denote it \overleftrightarrow{(ξ, η)}. This gives a one-to-one correspondence between a certain subset of PMF ×PMF and the set of geodesics in T .

Remark. It is not in general true that the end of the ray \overrightarrow{[σ, ξ)} converges in \overline{T} to the point ξ; however, it is at least true for cobounded rays.

Let T T denote the tangent space of T . There is an embedding T × PMF ↦ T T , taking (σ, ξ) to the tangent vector at σ of the ray \overrightarrow{[σ, ξ)} in T T , denoted D\overrightarrow{[σ, ξ)}. The image of this embedding will be denoted T^{1}T , called the unit tangent bundle of T , and T^{1}T is, in fact, a topological sphere bundle over T . Moreover, the map U_{0} → T^{1}T taking (ξ, η) to D\overrightarrow{[σ(ξ, η), η)}, is a homeomorphism. See [GM91] and [HM79] for proofs.

Remark. The Teichmüller metric is not a Riemannian metric, although it is a Finsler metric. As such, the unit tangent sphere T_{σ}^{1}T at each σ ∈ T is not a true ellipsoid in the vector space T_{σ}T , but instead a more general convex, centrally symmetric sphere [Roy71].

The flow on U_{0} defined by (ξ, η) · t = (e^{t}ξ, e^{−t}η) pushes forward under the homeomorphism U_{0} → T^{1}T to a flow on T^{1}T , namely the geodesic flow of T . In other words, each tangent vector v ∈ T^{1}T is tangent to a unique geodesic γ, and the geodesic flow v · t is obtained by pushing v forward a distance t along γ.

2.4 Pseudo-Anosov homeomorphisms

A homeomorphism h: Σ → Σ is pseudo-Anosov if there exists a transverse pair of measured foliations F_{x}, F_{y} and a λ > 1 such that h(F_{x}) = λF_{x} and h(F_{y}) = λ^{−1}F_{y}; the foliations F_{x}, F_{y} are called the stable and unstable measured foliations of h, and λ is the expansion factor of h. A mapping class H ∈ M is said to be pseudo-Anosov if and only if it has a representative h: Σ → Σ which is pseudo-Anosov.

By construction, a mapping class H ∈ M = Isom(T ) is pseudo-Anosov if and only if there exists a geodesic γ in T such that H(γ) = γ and the action of H on γ is a translation of nonzero length. In this case, the geodesic γ is unique and is called the axis of H, denoted Axis(H). Moreover, if h is a pseudo-Anosov homeomorphism representing H, with stable and unstable foliations F_{x}, F_{y} and expansion factor λ, then Axis(H) = γ(F_{x}, F_{y}) and the translation length of H equals log(λ).

Note that by a theorem of Bers [Ber78], a mapping class H ∈ M is pseudo-Anosov if and only if the function σ ↦ d(H, Hσ) has a positive minumum in T ; moreover this minimum is acheived precisely on Axis(H).

2.5 Quasi-isometries

Given K ≥ 1, C ≥ 0, a (K, C) quasi-isometry between metric spaces is a map f: X → Y such that:

(1) For all x_{1}, x_{2} ∈ X we have

\frac{1}{K} d_{X}(x_{1}, x_{2}) − C ≤ d_{Y} (f(x_{1}), f(x_{2})) ≤ Kd_{X}(x_{1}, x_{2}) + C

(2) d_{Y} (y, f(X)) ≤ C for each y ∈ Y.

If f satisfies (1) but not necessarily (2) then it is called a (K, C) quasi-isometric embedding. A quasi-isometric embedding f : R → X is a quasigeodesic in X.

A coarse inverse of a quasi-isometry f: X → Y is a quasi-isometry g: Y → X such that, for some constant C′ > 0, we have d(g∘f(x), x) < C′ and d(f ∘g(y), y) < C′ for all x ∈ X and y ∈ Y. Every (K, C) quasi-isometry f: X → Y has a K, C′ coarse inverse g: Y → X, where C′ depends only on K, C: for each y ∈ Y define g(y) to be any point x ∈ X such that d(f(x), y) ≤ C.

A fundamental fact observed by Efremovich, by Milnor [Mil68], and by Švarc, which we use repeatedly without mention, states that if a group G acts properly discontinuously and cocompactly by isometries on a proper geodesic metric space X, then G is finitely generated, and X is quasi-isometric to G equipped with the word metric.

Given a metric space X and A ⊂ X, we denote Nbhd_{r}(A) = {x ∈ X | d(x, r) ≤ A}, and given A, B ⊂ X, we denote the Hausdorff distance by

d_{H}(A, B) = inf{r ∈ [0, ∞] | A ⊂ Nbhd_{r}(B) and B ⊂ Nbhd_{r}(A)}

Given a metric space X, the self quasi-isometries of X are denoted Q̂I(X). Define the coarse equivalence relation on Q̂I(X) where f, g ∈ Q̂I(X) are coarsely equivalent, denoted f ≈_{c} g, if

sup_{x∈X} d(fx, gx) = C < ∞

We call C the coarseness constant. Composition of elements of Q̂I(X) gives a well-defined binary operation on the set of coarse equivalence classes of self quasiisometries of X, defining a group QI(X), the quasi-isometry group of X. If h: X → Y is a quasi-isometry of metric spaces then h induces an isomorphism QI(X) → QI(Y). In particular, when Γ is a finitely generated group the identity map is a quasi-isometry with respect to the word metrics of any two finite generating sets, and so the quasi-isometry group QI(Γ) is independent of choice of word metric on Γ.

A quasi-action of a group G on a metric space X is a map G×X → X, denoted (g, x) ↦ g · x, such that for some K ≥ 1, C ≥ 0 we have:

• For each g ∈ G the map x ↦ g · x is a (K, C)-quasi-isometry.

• For each g, h ∈ G, x ∈ X we have d(gh · x, g · (h · x)) ≤ C; in other words, L_{gh} ≈_{c} L_{g} ∘L_{h} with coarseness constant independent of g, h, where L_{γ} means left multiplication by γ.

The quasi-action is cobounded if there exists a bounded subset D having nonempty intersection with every orbit of the quasi-action. The quasi-action is proper if for each R > 0 there exists an integer m ≥ 0 such that for any x, y ∈ X the cardinality of the set {g ∈ G | (g · B(x, R)) ∩ B(y, R) ≠ ∅} is at most m.

A fundamental principle of geometric group theory says that if a finitely generated group G is quasi-isometric to a metric space X, then the left action of G on itself, when conjugated by a quasi-isometry G → X, defines a cobounded, proper quasi-action of G on X. To be precise, if we have coarsely inverse quasi-isometries h: X → G, h̄: G → X, then the formula g · x = h̄(gh(x)) defines a cobounded, proper quasi-action of G on X.

3 Schottky groups on Teichmüller space

In this section we recall from [FM00b] the motivation for and definition of Schottky subgroups of mapping class groups; see that paper for details and proofs.

Recall that a Schottky group in Isom(H^{n}) is a discrete, free use convex cocompact terminology subgroup F such that every orbit is quasiconvex in Isom(H^{n}). Equivalently, letting Λ be the limit set of F and HΛ the convex hull of Λ, the action of F on HΛ is cocompact; it follows that HΛ is quasi-isometric to F, and this quasi-isometry extends continuously to an F -equivariant homeomorphism between Λ and the Gromov boundary of F. This equivalence follows from the fact that H^{n} itself is a δ-hyperbolic metric space, and in fact the same (or closely analogous) equivalence holds for free subgroups of word hyperbolic groups, thereby providing a theory of Schottky subgroups of word hyperbolic groups.

In [FM00b] we mimic this setup for free subgroups F of Isom(T ) = M. The tricky part is that Teichmüller space T is not δ-hyperbolic. Nevertheless, when the properties above for Schottky subgroups of Isom(H^{n}) are carefully translated into the language of Isom(T ), the results of Minsky [Min96] provide enough negative curvature in T to prove the equivalence of various notions of Schottkiness for F. In addition, one of the main theorems of [FM00b] is that the Schottky condition characterizes precisely those free subgroups F for which π_{1}(Σ) ⋊ F is word hyperbolic.

Theorem 3.1 (Schottky groups: Definitions). Let F be a finite rank free subgroup of M(Σ). The following are equivalent:

1. Orbit quasiconvexity Each orbit O of the action of F on T is quasiconvex in T , i.e. there is a constant A such that for any x, y ∈ O, the geodesic \overline{xy} is contained in the A-neighborhood of O.

2. Weak convex cocompactness There is a continuous, F -equivariant embedding of the Gromov boundary ∂F into PMF, with image denoted Λ, satisfying the following:

(1) For any ξ ≠ η ∈ Λ there is a geodesic \overleftrightarrow{(ξ, η)} in T ; let

HΛ = ∪\{\overleftrightarrow{(ξ, η)} | ξ ≠ η ∈ Λ\}

be the weak convex hull of Λ, and let \overline{H}Λ = HΛ ∪ Λ.

(2) The F -equivariant homeomorphism Λ → ∂F extends to an F -equivariant

map

(\overline{H}Λ, Λ, HΛ) → (F ∪ ∂F, ∂F, F)

which is continuous at each point of Λ and which restricts to a quasiisometry HΛ → F, with respect to the Teichmüller metric on HΛ and the word metric on F.

3. Word hyperbolic extension The extension group Γ_{F} = π_{1}(Σ) ⋊ F is word hyperbolic.

Remark. It follows from the weak convex cocompactness property that for each geodesic \overleftrightarrow{(ξ, η)} in HΛ, the image of \overleftrightarrow{(ξ, η)} in F is a quasigeodesic whose ends converge in F ∪ ∂F to the images of ξ, η respectively. It also follows that each nontrivial element f ∈ F is pseudo-Anosov, because f has an axis in the Cayley graph of F and so f has an axis in T .

Theorem 3.1 is proved in [FM00b]. We call a subgroup F satisfying any one of the equivalent conditions of Theorem 3.1 a Schottky subgroup of M(Σ), or a Schottky group of mapping classes. These groups exist in abundance:

Theorem 3.2 (Abundance of Schottky groups). Let ϕ_{1}, …, ϕ_{n} be a collection of n independent pseudo-Anosov elements of M(Σ). Then for any sufficiently large natural numbers a_{1}, …, a_{n}, the subgroup of M(Σ) generated by ϕ_{1}^{a_{1}}, …, ϕ_{n}^{a_{n}} is a Schottky subgroup.

Proof. As noted in [FM00b], this follows from Theorem 3.1 together with the main result of [Mos97]. ♢

4 The geometry and topology of Γ_{H}

4.1 A geometric model for Γ_{H}

Let H be a Schottky subgroup of M = Isom(T ). We now build a contractible, piecewise-Riemannian 3-complex X_{H} on which Γ_{H} acts freely, properly discontinuously, and cocompactly by isometries, so that Γ_{H} is quasi-isometric to X.

Choose a free generating set h_{1}, …, h_{n} ∈ M for H. Let Δ be a graph with n edges δ_{1}, …, δ_{n}, each with one end at a common vertex v_{0} of valence n, and with opposite ends at valence one vertices v_{1}, …, v_{n}, respectively. Let R be the rose with n petals obtained from Δ by identifying v_{i} with v_{0} for each i = 1, …, n, and identify π_{1}(R, v_{0}) with H so that the homotopy class of the loop [δ_{i}] is identified with h_{i}.

On the product Σ × Δ make the following identifications: for each i choose a homeomorphism η_{i}: Σ → Σ representing h_{i} and identify Σ × v_{n} with Σ × v_{0} by identifying (x, v_{0}) ∼ (η_{i}(x), v_{i}) for each x ∈ Σ. Let K_{H} be the quotient 3-complex, and so we obtain a locally trivial fiber bundle K_{H} → R with fiber Σ and with monodromy H. Up to a bundle isomorphism homotopic to the identity, K_{H} does not depend on the choices of representatives η_{i}. By Van Kampen’s Theorem we have an isomorphism π_{1}(K_{H}) ≈ Γ_{H}.

Let X_{H} be the universal cover of K_{H}. Hence Γ_{H} acts properly discontinuously and cocompactly on X_{H}, with quotient K_{H}. We now specify a metric on X_{H} for which this action is isometric.

Let T_{H} be the universal cover of R, an infinite, regular, 2n-valent tree, regarded as the Cayley graph for H. The universal cover of Σ is the Poincaré disc D. The bundle K_{H} → R with fiber Σ lifts to a locally trivial fiber bundle π: X_{H} → T_{H} with fiber D, and it follows that X_{H} is homeomorphic to D × T_{H}. In order to construct a Γ_{H}-equivariant metric on X_{H} the metrics on the fibers of π must be appropriately “twisted” by considering the action of H on T .

We can embed the graph Δ in the tree T_{H} as a fundamental domain for the action of H, so that the restriction of the universal covering map T_{H} → R agrees with the quotient map Δ → R. Pick a base point σ_{0} in the Teichmüller space T . Choose a map ρ: Δ → T taking v_{0} to σ_{0}, taking v_{i} to h_{i}(σ_{0}), and taking δ_{i} to a smooth path between σ_{0} and h_{i}(σ_{0}), say the Teichmüller geodesic. Extend H-equivariantly to obtain a map ρ: T_{H} → T . Henceforth we shall fix the map ρ, and arcs in the tree T_{H} will be parameterized by arc length in T with respect to the map ρ. We use τ as a variable taking values in T_{H} and dτ as the arc length parameter in T_{H}.

Given an arc α ⊂ T_{H}, we can impose a hyperbolic structure on leaves of Σ×α in the form of a Riemannian metric g_{τ} on each Σ×τ, so that the conformal class of g_{τ} represents the point τ ∈ T , and so that the metrics g_{τ} vary smoothly with τ ∈ α. This can be obtained for instance by choosing a pair-of-pants decomposition of Σ and using the associated Fenchel-Nielsen coordinates for T (see [FM00b] for a further discussion). Then we may extend to obtain a piecewise smooth Riemannian metric on Σ × α by the formula

ds^{2} = g_{τ}^{2} + dτ^{2}

This metric is smooth on Σ × α except over the vertices of T_{H} in α.

Applying this to each edge δ_{i} of Δ we obtain hyperbolic structures on the fibers of Σ × Δ, varying smoothly over each edge δ_{i}, so that for each τ ∈ Δ the hyperbolic surface Σ × τ represents ρ(τ) ∈ T . We therefore obtain a piecewise smooth Riemannian metric on the 3-complex Σ × Δ. Since h_{i}(v_{0}) = v_{i} it follows that there is a unique isometry η_{i}: Σ×v_{0} → Σ×v_{i} representing the mapping class h_{i}, and we use these choices of η_{i} to construct K_{H} (here, for each τ ∈ Δ, we are implicitly identifying Σ×τ with Σ by projection to the first factor). Thus we have defined a piecewise smooth Riemannian metric on K_{H} whose restriction to each fiber of the bundle K_{H} → R is a hyperbolic metric on that fiber.

Lifting the metric on K_{H} we obtain a piecewise Riemannian metric on the universal cover X_{H}, whose restrictions to the fibers of the bundle π: X_{H} → T_{H} give a continuously varying family of hyperbolic metrics \overline{g}_{τ} on the fibers, parameterized by τ ∈ T_{H}. Each fiber is isometric to H^{2}. For each arc α of T_{H} the sub-bundle π^{−1}(α) has the form D × α and the Riemannian metric has the form

d\overline{s}^{2} = \overline{g}_{τ}^{2} + dτ^{2}

where \overline{g}_{τ} is the lift to D × τ of the metric g_{τ} on the appropriate fiber of K_{H}.

With respect to the piecewise Riemannian metric and the associated geodesic metric on X_{H}, the group Γ_{H} acts properly discontinuously and cocompactly by isometries.

4.2 What it means to coarsely respect a pattern

A pattern in a metric space is simply a collection of subsets. We will loosely use the term foliation to refer to a pattern forming a partition of the space into disjoint subsets called leaves, and in that context the pattern itself will be called the leaf space.

Let X, Y be metric spaces, and let F, G be patterns in X, Y respectively. A quasi-isometry ϕ: X → Y is said to coarsely respect the patterns F, G if there exists a number A ≥ 0 and a map h: F → G such that for each element L ∈ F we have

d_{H}(ϕ(L), h(L)) ≤ A

and if further a similar statement holds for a coarse inverse of ϕ. When distinct elements of F have infinite Hausdorff distance in X, and similarly for G, then h is a bijection and it is uniquely determined by the quasi-isometry ϕ.

An isometry of X_{H} respects many different patterns in X_{H}. The idea of the proof of Theorem 1.3 is to show that an arbitrary quasi-isometry f of Γ_{H} (hence of X_{H}) preserves more and more structure, in the sense of coarsely respecting finer and finer patterns, until so much structure is preserved that f must actually be a bounded distance from an element of the extension group Γ_{C} defined in Theorem 1.3.

4.3 Hyperplanes and the horizontal foliation

The bundle π: X_{H} → T_{H} associated to the group Γ_{H} determines two important patterns in X_{H}: the horizontal foliation and the pattern of hyperplanes.

The horizontal foliation of X_{H} is the pattern of fibers of π, subsets D_{τ} = π^{−1}(τ), τ ∈ T_{H}, each called a horizontal leaf of X_{H}. The leaf space of the horizontal foliation is identified with T_{H} via the bundle map π. Any two horizontal leaves have finite Hausdorff distance in X_{H}, in fact from the form of the metric on X_{H} we see easily that d_{H}(D_{τ} , D_{u}) = d(τ, u) for all τ, u ∈ T_{H}. In other words, the bundle map π induces an isometry between the horizontal foliation equipped with the Hausdorff metric and the metric tree T_{H}. As noted earlier, the metric on X_{H} restricts to a metric \overline{g}_{τ} on each horizontal leaf D_{τ} making D_{τ} isometric to H^{2}.

A hyperplane in X_{H} is any set P_{w} = π^{−1}(w), where w is a bi-infinite geodesic in T_{H}. The metric on X_{H} restricts to a piecewise Riemannian metric on P_{w} of the form g̃_{τ}^{2} + dτ^{2}, with respect to the coordinates P_{w} ≈ D × w. Any two distinct hyperplanes in X_{H} have infinite Hausdorff distance.

The pattern of hyperplanes in X_{H} is the first pattern which must be coarsely respected by a quasi-isometry.

Proposition 4.1 (Hyperplanes respected). Any quasi-isometry f: X_{H} → X_{H} coarsely respects the pattern of hyperplanes in X_{H}, with coarseness constant depending only on the quasi-isometry constants of f.

In other words, f must map each hyperplane in X_{H} a uniform Hausdorff distance from some other hyperplane in X_{H}. Since distinct hyperplanes have infinite Hausdorff distance in X_{H} it follows that f induces a bijection on the pattern of hyperplanes; when f is understood we denote this bijection P_{w} ↦ P_{w′}.

Each hyperplane P_{w} is uniformly properly embedded in X_{H}, that is, there is a proper function r: [0, ∞) → [0, ∞) independent of w such that for all x, y ∈ P_{w} we have:

d_{X_{H}} (x, y) ≥ r(d_{P_{w}}(x, y))

By applying Proposition 4.1 together with a simple general principle (made explicit in Lemma 2.1 of [FM00a]) we have for any quasi-isometry f : X_{H} → X_{H} the following fact: for each hyperplane P_{w}, the map f induces via restriction composed with nearest-point projection, a quasi-isometry ϕ: P_{w} → P_{w′}; the quasi-isometry constants for ϕ depend only on those for f.

The horizontal foliation on X_{H} restricts to a horizontal foliation on P_{w}. Using the coordinates P_{w} ≈ D × R as described above, the horizontal leaves in P_{w} are of the form D_{t} = D × t, t ∈ R. The leaf space of this foliation is R and we have d_{H}(D_{s}, D_{t}) = |s − t| using Hausdorff distance in P_{w}.

The fact that X_{H} fibers over a tree T_{H} with nontrivial branching restricts the behavior of ϕ as follows:

Proposition 4.2 (Horizontal foliation respected). For any quasi-isometry f: X_{H} → X_{H} and any hyperplane P_{w}, the induced quasi-isometry ϕ: P_{w} → P_{w′} uniformly coarsely respects horizontal foliations.

In other words, there exists a constant A independent of w such that for any horizontal leaf D_{t} ⊂ P_{w} there is a horizontal leaf D_{t′} ⊂ P_{w′} with d_{H}(ϕ(D_{t}), D_{t′}) ≤ A

This is precisely where the assumption is used that H has rank greater than one.

Proof of Propositions 4.1 and 4.2. Since H has rank greater than one, its Cayley graph T_{H} is a bushy tree, meaning that each point of T_{H} is within some fixed distance β = 1 of some vertex v such that T − v has at least 3 unbounded components. We can thus apply the following result, which is Theorem 7.7 of [FM00a], to the metric fibration π: X_{H} → T_{H}.

Lemma 4.3. Let π: X → T, π′: X′ → T′ be metric fibrations over bushy trees T, T′, such that the fibers of π and π′ are contractible n-manifolds for some n. Let f: X → X′ be a quasi-isometry. Then there exists a constant A, depending only on the metric fibration data of π, π′, the quasi-isometry data for f, and the constant β, such that:

(1) For each hyperplane P ⊂ X there exists a unique hyperplane Q ⊂ X′ such that d_{H}(f(P), Q) ≤ A.

(2) For each horizontal leaf L ⊂ X there is a horizontal leaf L′ ⊂ X′ such that d_{H}(f(L), L′) ≤ A.

Proposition 4.1 is an immediate consequence. To obtain Proposition 4.2, consider a horizontal leaf D_{t} of P_{w}. By applying Lemma 4.3 we obtain a horizontal leaf D_{s} of X_{H} uniformly Hausdorff close to f(D_{t}). But f(P_{w}) is uniformly Hausdorff close to P_{w′} and so D_{s} is uniformly Hausdorff close to some horizontal leaf D_{t′} ⊂ P_{w′}. Finally, the closest point projection f(P_{w}) ↦ P_{w′} moves points a uniformly bounded distance, and so f(P_{w}) is a uniformly Hausdorff close to its closest point projection in P_{w′}. ♢

4.4 The singular solv metric on a hyperplane

In this subsection we find a different metric on hyperplanes P_{w} that will allow us to apply ideas from pseudo-Anosov dynamics.

Associated to each geodesic γ in T we recall the construction of a “singular solv metric” on D × γ. Choose a normalized transverse pair of measured foliations F_{x}, F_{y} on Σ, with transverse measures |dy|, |dx| respectively, so that γ = γ(F_{x}, F_{y}); we can parameterize γ by arc length:

γ(t) = σ(e^{t}F_{x}, e^{−t}F_{y})

For each t we have a singular Euclidean metric μ_{t} = μ(e^{t}F_{x}, e^{−t}F_{y}), that is,

dμ_{t}^{2} = e^{−2t}dx^{2} + e^{2t}dy^{2}

Note that the transverse pair (e^{t}F_{x}, e^{−t}F_{y}) is normalized for all t, that is, Area(μ_{t}) = ∫_{Σ} e^{t}|dy|e^{−t}|dx| = ∫_{Σ} |dx| |dy| = 1. Lifting to the universal cover D = Σ̃ we obtain a transverse pair of measured foliations F̃_{x}, F̃_{y} with transverse measures still denoted |dy|, |dx|, and a one-parameter family of singular Euclidean metrics μ̃_{t} on D, given by

dμ̃_{t}^{2} = e^{−2t}dx^{2} + e^{2t}dy^{2}

Now we can define a singular Riemannian metric on D × γ by the formula

ds^{2} = dμ̃_{t}^{2} + dt^{2}
= e^{−2t}dx^{2} + e^{2t}dy^{2} + dt^{2}

The singular locus of this metric is a family of vertical lines in D × γ, whose intersection with the fiber D × t is precisely the singular set of e^{t}F̃_{x}, e^{−t}F̃_{y}. The Riemannian metric extends across the singular locus to a complete geodesic metric on D × γ. The space D × γ with this metric is called the singular solv space associated to γ, denoted Q_{γ}. The reason for the terminology is that away from the singular locus the metric is locally modelled on 3-dimensional solv-geometry, and at the singular locus (each component of which is a line) the metric is modelled on some number of “half solv’s” glued together.

Note that the singular solv-metric on Q_{γ} is uniquely determined by the fibration Q_{γ} → γ together with the family of xy-structures e^{t}F̃_{x}, e^{−t}F̃_{y}: in each leaf; the measured foliation e^{t}F̃_{x} provides the e^{2t}dy term and the measured foliation e^{−t}F̃_{y} provides the e^{−2t}dx term, and the map to γ with the arc length parameter t provides the dt^{2} term.

In summary, we have a bijective correspondence

{geodesics γ in T } ↔ {π_{1}(Σ)-equivariant singular solv metrics on D × R}

xy-structures. For convenience we shall define an xy-structure on a surface S to be a transverse pair of measured foliations F_{x}, F_{y}. We will work with xy-structures on Σ as well as the lifted structures on the universal cover D.

An affine automorphism of an xy-structure on S is a homeomorphism of S which respects the (unordered) pair of (unmeasured) foliations {F_{x}, F_{y}}, multiplying the transverse measure on one by λ > 0 and multiplying the transverse measure on the other by 1/λ. Let Aff(F_{x}, F_{y}) be the group of affine automorphisms of the xy-structure F_{x}, F_{y}. There is a subgroup of index ≤ 2 in Aff(F_{x}, F_{y}), denoted Aff_{+}(F_{x}, F_{y}), which respects the ordered pair of (unmeasured) measured foliations (F_{x}, F_{y}), with transverse measures multiplied as described above. There is a homomorphism

Stretch: Aff(F_{x}, F_{y}) → SL(2, R)

whose image lies in the subgroup of SL(2, R) corresponding to matrices which are either zero off the diagonal or zero on the diagonal, as follows. Given ϕ ∈ Aff(F_{x}, F_{y}): if ϕ(F_{x}) = λF_{x} and ϕ(F_{y}) = \frac{1}{λ}F_{y} then Stretch(ϕ) is the matrix ( \begin{matrix}λ & 0 \\ 0 & 1/λ \end{matrix} ); whereas if ϕ(F_{x}) = λF_{y} and ϕ(F_{y}) = \frac{1}{λ}F_{x} then Stretch(ϕ) is the matrix ( \begin{matrix}0 & λ \\ 1/λ & 0 \end{matrix} ). In the cases of interest where S = Σ or D, the image of the homomorphism, denoted Stretch(F_{x}, F_{y}), is a discrete subgroup of SL(2, R); discreteness follows easily using the fact that the singular set of the pair F_{x}, F_{y} forms a nonempty, discrete net in S. As a consequence, Stretch(F_{x}, F_{y}) is isomorphic to either D_{∞}, Z, Z/2, or the trivial group. The kernel of the stretch homomorphism is the group Isom_{+}(F_{x}, F_{y}) of automorphisms of the ordered pair of measured foliations F_{x}, F_{y}.

We obtain a short exact sequence

1 → Isom_{+}(F_{x}, F_{y}) → Aff(F_{x}, F_{y}) → Stretch(F_{x}, F_{y}) → 1

where Isom_{+}(F_{x}, F_{y}) is the group of automorphisms of the ordered pair of measured foliations F_{x}, F_{y}.

Given a geodesic γ = γ(F_{x}, F_{y}) in T , the universal cover D = Σ̃ with the lifted xy-structure F̃_{x}, F̃_{y} may be identified with a certain horizontal fiber of Q_{γ}.

With respect to this identification, every isometry of the singular solv-manifold Q_{γ} respects the horizontal foliation, and vertical projection of an isometry onto D defines an xy-affine automorphism of F̃_{x}, F̃_{y}. Every xy-affine automorphism arises in this manner, leading to an isomorphism between Isom(Q_{γ}) and Aff(F̃_{x}, F̃_{y}). This leads in turn to an isomorphism of short exact sequences as follows:

[figure: 1; Isom_{+}(F̃_{x}, F̃_{y}); Aff(F̃_{x}, F̃_{y}); Stretch(F̃_{x}, F̃_{y}); 1; 1; Isom_{h}(Q_{γ}); Isom(Q_{γ}); C_{γ}; 1]

where Isom_{h}(Q_{γ}) is the subgroup of Isom(Q_{γ}) preserving each horizontal leaf, and C_{γ} = Isom(Q_{γ})/ Isom_{h}(Q_{γ}); the latter group is isomorphic to D_{∞}, Z, Z/2, or the trivial group.

We also have an isomorphism of short exact sequences

[figure: 1; Isom_{+}(F̃_{x}, F̃_{y}); Aff_{+}(F̃_{x}, F̃_{y}); Stretch_{+}(F̃_{x}, F̃_{y}); 1; 1; Isom_{h}(Q_{γ}); Isom_{+}(Q_{γ}); C_{γ+}; 1]

where Aff_{+}(F̃_{x}, F̃_{y}) was defined earlier, Stretch_{+}(F̃_{x}, F̃_{y}) is the intersection of Stretch(F̃_{x}, F̃_{y}) ⊂ SL(2, R) with the diagonal subgroup of SL(2, R), Isom_{+}(Q_{γ}) is the subgroup of Isom(Q_{γ}) preserving the transverse orientation on the horizontal foliation, and C_{γ+} = Isom_{+}(Q_{γ})/ Isom_{h}(Q_{γ}); in each of these four cases, the + subscript induces a subgroup of index ≤ 2. The group Stretch_{+}(F̃_{x}, F̃_{y}) ≈ C_{γ+} is either Z or trivial.

Note that C_{γ} contains the group Stab_{T} (γ) = {Φ ∈ Isom(T ) | Φ(γ) = γ}, and we shall show in Lemma 6.2 that this containment has finite index.

Schottky groups and singular solv-manifolds. Now consider a Schottky group H ⊂ M with limit set Λ ⊂ PMF, and weak convex hull HΛ. We have the Cayley graph T_{H} and an H-equivariant immersion ρ: T_{H} → T ; let \overline{T}_{H} = T_{H} ∪ Λ. An immediate corollary of Theorem 3.1 is that T_{H} and HΛ have finite Hausdorff distance in T , and so we obtain an H-equivariant quasi-isometry θ: HΛ → T_{H} which moves points a uniformly bounded distance in T . It also follows from Theorem 3.1 that the map θ extends, via the identity map on Λ, to an H-equivariant map \overline{θ}: \overline{H}Λ → \overline{T}_{H} which is continuous at each point of Λ. It follows immediately that for any ξ ≠ η ∈ Λ, the restriction of θ to the geodesic \overleftrightarrow{(ξ, η)} under θ is a quasigeodesic, with quasigeodesic constants independent of ξ, η, and the ends of this quasigeodesic converge in \overline{T}_{H} to ξ, η respectively; the unique geodesic in T_{H} with these endpoints is denoted \overline{ξη}. To summarize:

• The correspondence between geodesics γ = \overleftrightarrow{(ξ, η)} in HΛ and geodesics w = \overline{ξη} in T_{H} is a bijection. We denote this correspondence by w = w_{γ}, γ = γ_{w}. Corresponding geodesics γ, w_{γ} are uniformly Hausdorff close in T .

We need to be a bit more precise. By lifting each geodesic in HΛ to the unit tangent bundle T^{1}T we get a closed subset of T^{1}T invariant under the geodesic flow denoted T HΛ. The space T HΛ is locally homeomorphic to Λ × Λ × R, i.e. it is locally a Cantor set crossed with the line. We will often confuse a geodesic γ in HΛ with its lift to T HΛ; these geodesics form a lamination of T HΛ. As we have said, the map θ: HΛ → T_{H} lifts to a map from T HΛ to T_{H}, taking each geodesic γ to a quasigeodesic in T_{H} uniformly Hausdorff close to w_{γ}. But then, by moving the map T HΛ → T_{H} a bounded amount, we obtain a continuous map Θ: T HΛ → T_{H} with the property that Θ(γ) = w_{γ} for each γ. The restriction to γ is denoted Θ_{γ}: γ → w_{γ}, and this is a quasi-isometry with constants independent of γ, and we may take Θ_{γ} to be a homeomorphism.

Proposition 4.4 (Comparison of metrics). Given corresponding geodesics w in T_{H} and γ = γ_{w} in T HΛ, there exists a π_{1}Σ-equivariant, horizontal respecting quasi-isometry F_{w}: P_{w} → Q_{γ}, with quasi-isometry constants independent of w, such that F_{w} is a lift of the map Θ_{γ}^{−1} : w → γ.

In other words, the natural metric on each hyperplane of the geometric model space X_{H} is uniformly quasi-isometric to a singular solv 3-manifold. This is the key place where we use the fact that H is a Schottky group, and not just any free group of pseudo-Anosovs.

Proof. Recall that we have a Γ_{H}-equivariant D-bundle X_{H} → T_{H}, carrying a Γ_{H}-equivariant piecewise Riemannian metric whose restriction to each fiber is isometric to H^{2}, with Γ_{H} acting cocompactly. Each of the spaces P_{w} is embedded in X_{H} as the inverse image of w ⊂ T_{H}.

We now define a Γ_{H}-equivariant D-bundle Ξ → T HΛ in which each of the singular solv-manifolds Q_{γ} sits, as follows.

First, note that each point of T^{1}T corresponds to (the isotopy class of) a normalized xy-structure (F_{x}, F_{y}) on Σ. We may assemble these structures into an M-equivariant Σ-bundle over T^{1}T , each fiber equipped with an xy-structure in the appropriate isotopy class, so that the xy-strucures vary continuously as the base point in T^{1}T varies. Restricting to T HΛ we obtain an H-equivariant xy-Σ-bundle Υ → T HΛ.

The universal cover of the Σ-bundle over T^{1}T is a D-bundle over T , with smoothly varying xy-structures on the fibers, on which the π_{1}Σ extension of M acts, namely the once-punctured mapping class group M(Σ, p). Restricting to T HΛ we obtain a Γ_{H}-equivariant xy-D-bundle Ξ → T HΛ. Restricting to any geodesic γ ⊂ T HΛ we obtain the fibration Q_{γ} → γ. Note that the foliation of T HΛ by geodesics lifts to a foliation of Ξ by 3-manifolds: the 3-manifold over the geodesic γ ⊂ T HΛ is Q_{γ}. The fiberwise xy-structures vary continuously in Ξ, and the arc length parameter on geodesics of T HΛ varies continuously; as noted earlier, these data determine singular solv-metrics on each Q_{γ}, and these metrics vary continuously in Ξ.

Now lift the H-equivariant map Θ: T HΛ → T_{H} to a Γ_{H}-equivariant continuous map of D-bundles, Θ̃: Ξ → X_{H}, taking each fiber of Ξ homeomorphically to the corresponding fiber of X_{H}, and taking each singular solv manifold Q_{γ} to the corresponding singular Riemannian manifold P_{w_{γ}}.

By cocompactness of the Γ_{H} actions on Ξ and X_{H}, and by continuity of the xy-structures on fibers of Ξ and the H^{2} structures on fibers of X_{H}, it follows that Θ̃ induces quasi-isometries from Ξ fibers to X_{H} fibers with uniform quasi-isometry constants; in particular, we get uniform quasi-isometries from the horizontal sets of Q_{γ} to the horizontal sets of P_{w_{γ}}, over all geodesics γ in T HΛ. Moreover, since the map Θ: T HΛ → T_{H} is uniformly quasi-isometric from a geodesic γ in T HΛ to the corresponding geodesic w_{γ} in T_{H}, it follows that Θ̃ induces a uniform quasiisometry from the |dt| term in the metric on Q_{γ} to the dτ term in the metric on P_{w_{γ}}. This implies that the family of maps Q_{γ} → P_{w_{γ}} is uniformly quasi-isometric. ♢

5 Quasi-isometries remember the dynamics

Let H ⊂ M be a Schottky subgroup and let ϕ: Γ_{H} → Γ_{H} be a quasi-isometry. By Propositions 4.1, 4.2, and 4.4, to each bi-infinite path w ∈ T_{H} there corresponds a bi-infinite path w′ ∈ T′_{H}, and a horizontal respecting quasi-isometry Q_{γ_{w}} → Q_{γ_{w′}} of singular solv spaces. In this section we study such quasi-isometries, and show that they must coarsely respect certain dynamically defined foliations.

5.1 The vertical flow on Q_{γ}

Let γ be a fixed Teichmüller geodesic in T (Σ), and let Q_{γ} be the associated singular solv metric on D × γ ≈ D × R. There is a natural flow on Q_{γ} given by

Ψ_{s}(x, t) = (x, t + s)

This flow is called the vertical flow, and its orbits are called vertical geodesics or vertical flow lines in Q_{γ}. This flow is a pseudo-Anosov flow in the sense of [FM00c]. The stable and unstable foliations of Ψ expand and contract at the uniform rate e^{t}. Singular orbits of Ψ are the same as singular vertical geodesics of Q_{γ}.

There are three naturally defined foliations of Q_{γ} which are invariant under the vertical flow: the codimension-2 foliation by vertical flow lines; the codimension-1 weak stable foliation; and the codimension-1 weak unstable foliation. Given a vertical flow line ℓ, the weak stable (resp. unstable) leaf containing ℓ is the union of vertical flow lines that asymptote to ℓ as t → ∞ (resp. t → −∞). Each of the weak stable and unstable foliations has some nonmanifold leaves, one such leaf containing each singular vertical geodesic. Note that for each leaf L of the weak stable or unstable foliation, if L is nonsingular then L is isometric to a hyperbolic plane, whereas if L is the singular leaf through a singular vertical geodesic ℓ then L is isometric to a union of hyperbolic half-planes meeting along their common boundary ℓ; in either case, L can be expressed as a finite union of hyperbolic planes.

Our goal now is to use pseudo-Anosov dynamics to show that a horizontalrespecting quasi-isometry between two hyperplanes must also coarsely respect all of these foliations. In short: the quasi-isometry remembers the dynamics.

5.2 Coarse intersection

We need a basic notion from coarse topology.

Definition (Coarse intersection). A subset W of a metric space X is a coarse intersection of subsets U, V ⊂ X, denoted W = U ∩_{c}V, if there exists C_{0} such that for every C ≥ C_{0} there exists A = A(C) ≥ 0 so that

d_{H}(Nbhd_{C}(U) ∩ Nbhd_{C}(V), W) ≤ A

Note that although such a set W may not exist, when it does exist then any two such sets are a bounded Hausdorff distance from each other. The function A(C), C ≥ C_{0} is called the coarse intersection function.

We will need the following fact, which is an elementary consequence of the definitions.

Lemma 5.1. For any quasi-isometry f: X → Y of metric spaces, and U, V ⊂ X, if U ∩_{c} V exists then f(U ∩_{c} V) is a coarse intersection of f(U), f(V), with coarse intersection function depending only on the quasi-isometry constants for f and the coarse intersection function for U and V. ♢

5.3 The dynamically defined foliations are respected

In this subsection we prove the key proposition:

Proposition 5.2 (Stable and unstable foliations respected). Let γ, γ′ be cobounded Teichmüller geodesics. Then any horizontal-respecting quasi-isometry ϕ: Q_{γ} → Q_{γ′} coarsely respects the stable and unstable foliations of the vertical flows on Q_{γ}, Q_{γ′}. Also, ϕ coarsely respects the patterns of singular stable and unstable leaves.

Every vertical flow line can be realized as the coarse intersection of a stable and unstable leaf, with uniform coarse intersection function independent of the choice of flow line. Proposition 5.2 and Lemma 5.1 therefore imply that ϕ coarsely respects the collection of vertical geodesics. A similar argument works for the collection of singular geodesics, using singular stable and unstable leaves. We record this as:

Corollary 5.3 (Vertical flow lines respected). The quasi-isometry ϕ of Proposition 5.2 coarsely respects the foliations of vertical flow lines in Q_{γ} and Q_{γ′}, as well as the patterns of singular vertical lines.

Proof of Proposition 5.2. The horizontal foliation of Q_{γ} is an example of a uniform foliation, which means that any two leaves have finite Hausdorff distance. Any map between two horizontal leaves which moves points a bounded distance in Q_{γ} is a quasi-isometry between those leaves. It follows that there is a canonical coarse equivalence class of quasi-isometries between any two leaves, and moreover the composition of two such quasi-isometries is another one. Each leaf is quasiisometric to the hyperbolic plane H^{2} and its Gromov boundary is a circle, so there is a canonical identification of all the circles at infinity to a single circle which we denote SQ_{γ}. These facts were noted by Thurston in [Thu97].

It is well-known that a quasi-isometry H^{2} → H^{2} induces a homeomorphism ∂H^{2} → ∂H^{2} between the circles at infinity, and that coarsely equivalent quasiisometries induce the same boundary map. Any horizontal respecting quasiisometry Q_{γ} → Q_{γ′} therefore induces a homeomorphism SQ_{γ} → SQ_{γ′} between their respective circles at infinity. The underlying idea of our proof is to find additional quasi-isometrically invariant structures on SQ_{γ} and SQ_{γ′} which encode the stable and unstable foliations, and use this information to prove quasi-isometric invariance of these foliations.

To proceed with the proof we need some notation. Let D_{t} denote the plane D × t ⊂ D × R = Q_{γ}. The plane D_{t} comes equipped with an xy-structure, as explained in §4.4. For each s, t let ϕ_{st}: D_{s} → D_{t} be the map (p, s) → (p, t), p ∈ D. In other words, ϕ_{st} flows along vertical flow segments of Q_{γ} from D_{s} to D_{t}; each such flow segment has length |s − t|, and so ϕ_{st} is a quasi-isometry in the canonical coarse equivalence class from D_{s} to D_{t} as discussed above. In fact, the map ϕ_{st} is an xy-affine homeomorphism with stretch factor e^{|s−t|}, implying that ϕ_{st} is e^{|s−t|}-bilipschitz. Note that ϕ_{tu} ∘ ϕ_{st} = ϕ_{su}, for all s, t, u ∈ R.

The xy-metric on D_{t} is a CAT(0) metric, and it is also Gromov hyperbolic because D_{t} is quasi-isometric to H^{2}. The boundary ∂D_{t} is a circle, and each quasi-isometry ϕ_{st}: D_{s} → D_{t} induces a homeomorphism ∂ϕ_{st}: ∂D_{s} → ∂D_{t}.

For any two points ξ, η ∈ ∂D_{t} there exists an xy-geodesic with endpoints ξ, η, and any two such geodesics are the boundary of an isometric embedding of R×[a, b] for some [a, b] ∈ R (see [BH99]). Since D_{t} is not the Euclidean plane but has a cocompact isometry group, there is an upper bound on b − a independent of ξ, η. This bound is moreover independent of t, because coboundedness of the Teichmüller geodesic γ and compactness of the fibers of T^{1}T together imply that the collection of locally CAT(0) metric spaces D_{t}/π_{1}(Σ) lies in a compact space of locally CAT(0) metrics (this is the one place in the proof where we use coboundedness of the Teichmüller geodesics in the weak convex hull of H). Let R_{0} be a t-independent bound for |b − a|.

For each bi-infinite geodesic θ on D_{0}, the set

V = V_{θ} = ⋃_{t∈R}{ϕ_{0t}(θ)}

is called a vertical plane in Q_{γ}. Note that if ∂θ = {ξ, η} ⊂ ∂D_{0} then ∂ϕ_{0t}(θ) = {∂ϕ_{0t}(ξ), ∂ϕ_{0t}(η)} ⊂ ∂D_{t}, and so to V there is associated a unique pair of points in the circle SQ_{γ} which we call the endpoints of V. Moreover, the discussion in the previous paragraph shows that the Hausdorff distance between any two vertical planes of Q_{γ} with the same endpoints ξ, η is at most R_{0}.

The first structure which ϕ: Q_{γ} → Q_{γ′} must coarsely respect is the collection of vertical planes.

Claim 5.4 (Vertical planes preserved). Let ϕ: Q_{γ} → Q_{γ′} be a horizontalrespecting quasi-isometry. If V is any vertical plane in Q_{γ}, then ϕ(V) is a bounded Hausdorff distance from some vertical plane V′ in Q_{γ′}.

To prove Claim 5.4, let t → t′ be a bijective quasi-isometry of R such that ϕ(D_{t}) is Hausdorff close to D′_{t′}, with a uniform Hausdorff constant; we assume that the parameterizations are chosen so that 0 → 0 under this quasi-isometry. Composing the map D_{t} → ϕ(D_{t}) with a uniformly finite distance map ϕ(D_{t}) → D′_{t′}, we obtain a quasi-isometry ψ_{t}: D_{t} → D′_{t′} whose quasi-isometry constants are independent of t.

Fix a vertical plane V = V_{θ} and for each s consider the geodesic θ_{s} = V ∩ D_{s}. Its image quasigeodesic ψ_{s}(θ_{s}) is uniformly Hausdorff close in D′_{s′} to some geodesic θ′_{s′}, in particular ϕ_{0}(θ_{0}) is close to θ′_{0}. We need only show that for each s, θ′_{s′} is uniformly Hausdorff close to ϕ′_{0s′}(θ′_{0}), for then we can set

V′ = ∪_{s′}{ϕ′_{0s′}(θ′_{0})}

and it follows that ϕ(V) is uniformly Hausdorff close to V′.

Since any two geodesics in D′_{s′} which are Hausdorff close are R_{0}-Hausdorff close, we need only show that the Hausdorff distance between θ′_{s′} and ϕ′_{0s′}(θ′_{0}) is finite, in other words these two geodesics have the same endpoints in ∂D′_{s′}. But this is an immediate consequence of the coarse commutativity of the following diagram:

\xymatrix{D_{0} \ar[r]^{ψ_{0}} \ar[d]_{ϕ_{0s}} & D′_{0} \ar[d]^{ϕ′_{0s′}} \\ D_{s} \ar[r]_{ψ_{s}} & D′_{s′}}

This finishes the proof of Claim 5.4. We now find finer structures which must be coarsely respected by ϕ.

Each vertical plane V comes equipped with a horizontal foliation, obtained by intersecting the plane with the horizontal foliation {D_{t}} of Q_{γ}. Denote the leaves of this foliation by

θ_{t} = D_{t} ∩ V, t ∈ R

so that θ_{t} = ϕ_{st}(θ_{s}) for any s, t ∈ R. The horizontal foliation {θ_{t}} has a transverse orientation, pointing in the direction of increasing t. Note that the Hausdorff distance in V between θ_{t} and θ_{s} is exactly |t − s|. There is a projection π: V → R with θ_{t} = π^{−1}(t). Define a quasivertical line in V to be a subset L ⊂ V such that the projection π: L → R is a quasi-isometry.

We divide the collection of vertical planes into three types: stable, unstable, and doubly unstable. A stable vertical plane is one which is contained in a leaf of the stable foliation on Q_{γ}; thus it is either a regular leaf of the stable foliation, or it is a union of two half-planes of a singular leaf. Similarly an unstable vertical plane is one contained in an unstable leaf. All other vertical planes are called doubly unstable vertical planes.

The three types of vertical planes—stable, unstable, and doubly unstable— can be distinguished from each other by horizontal respecting quasi-isometries which respect the transverse orientation, by observing the asymptotic behavior of quasivertical lines. To make this more precise, say that two quasivertical lines L, L′ ⊂ P are upward Hausdorff close if the Hausdorff distance between L ∩ π^{−1}[0, ∞) and L′∩π^{−1}[0, ∞) is finite; downward Hausdorff close is similarly defined using (−∞, 0].

Claim 5.5. Let V be a vertical plane. Then:

Stable: If V is a stable plane, then any two quasivertical lines in V are upward Hausdorff close but not downward Hausdorff close.

Unstable: If V is an unstable plane, then any two quasivertical lines in V are downward Hausdorff close but not upward Hausdorff close.

Doubly unstable: If V is a doubly unstable plane then there exist two quasivertical lines in V which are neither upward Hausdorff close nor downward Hausdorff close.

To prove Claim 5.5 when V is stable or unstable, observe first that V with its horizontal foliation {θ_{t}} is isometric to the hyperbolic plane H^{2} in the upper half plane model, with the “horizontal” horocyclic foliation centered on the point ∞. If V is stable (resp. unstable) then the transverse orientation points towards ∞ (resp. away). Next observe that any quasivertical line in H^{2} is a quasigeodesic with one endpoint at ∞, and so any two quasivertical lines are Hausdorff close in the direction of ∞.

To prove Claim 5.5 when V is doubly unstable, observe that θ_{0} = V ∩ D_{0} is an xy-geodesic in D_{0} which is not contained in a leaf of either the x-foliation or the y-foliation on D_{0}. There exists, therefore, two points p, q ∈ θ_{0} which do not lie in the same leaf of either the x-foliation or the y-foliation on D_{0}. The vertical flow lines p · R, q · R in Q_{γ} are evidently neither upward nor downward Hausdorff close in the singular solv-metric on Q_{γ}, and so the same is true in V. This proves Claim 5.5.

The same discussion holds, of course, in Q_{γ′}. Now consider a horizontalrespecting quasi-isometry ϕ: Q_{γ} → Q_{γ′}. By reversing the upward orientation in Q_{γ′} if necessary, we may assume that ϕ respects the upward orientation. Let V be any vertical plane in Q_{γ}. We have shown in Claim 5.4 that ϕ(V) is Hausdorff close to some vertical plane V′ in Q_{γ′}. Composing the map V → ϕ(V) with any finite distance map ϕ(V) → V′, we obtain therefore a quasi-isometry ψ: V → V′. Each horizontal leaf of V (resp. V′) is a coarse intersection of V with a horizontal leaf of Q_{γ} (resp. Q_{γ′}), and it follows that ψ coarsely respects the horizontal foliations and their transverse orientations.

Claim 5.5 shows manifestly that stable vertical planes are coarsely respected by ϕ: Q_{γ} → Q_{γ′}, and similarly for unstable vertical planes. To finish the proof of Proposition 5.2, given two stable (resp. unstable) vertical planes V_{1}, V_{2} in Q_{γ} or in Q_{γ′}, we must give a quasi-isometrically invariant property which characterizes V_{1}, V_{2} lying in the same weak stable (resp. unstable) leaf. Namely, V_{1}, V_{2} lie in the same leaf if and only if, for any t ∈ R, the triple coarse intersection V_{1} ∩_{c} V_{2} ∩_{c} D_{t} is unbounded. ♢ Remark. There is an alternative “dynamical” proof of Proposition 5.2 which we worked out first, and which parallels the proof of the analogous proposition in [FM00a]. The latter is concerned with the solvable Lie groups associated to geodesics in the symmetric space of GL(n, R), and the natural Anosov flows on these solvable Lie groups; the proof in [FM00a] is an easy consequence of the shadowing lemma for Anosov flows. Unfortunately the shadowing lemma is false in pseudo-Anosov dynamics, complicating the situation drastically. There is an alternative shadowing theory for pseudo-Anosov dynamical systems, developed in [Han85], [Han88], and [Mos89], and this was used in our original proof of Proposition 5.2. The proof given above for Proposition 5.2 entirely avoids these issues by using Gromov hyperbolicity of the horizontal leaves—a fact which was not available in [FM00a], where the horizontal leaves are Euclidean.

6 Periodic hyperplanes

A hyperplane P_{w} in X_{H} is a periodic hyperplane if w is a periodic geodesic in T_{H}, that is, the subgroup C_{w} of H stabilizing w is infinite cyclic. Periodic hyperplanes P_{w} are special in that they admit a priori extra isometries, above and beyond the isometric action of π_{1}(Σ): the subgroup π_{1}(Σ) ⋊ C_{w} of Γ_{H} acts isometrically on P_{w}.

Our goal now is to show that horizontal-respecting quasi-isometries of periodic hyperplanes must remember the extra symmetries.

Recall that w ↔ γ_{w} is a bijection between geodesics in the tree T_{H} and Teichmüller geodesics in HΛ.

Proposition 6.1 (Periodic hyperplanes preserved). Given w, w′ geodesics in T_{H}, suppose that there exists a horizontal respecting quasi-isometry between hyperplanes ϕ: P_{w} → P_{w′}. Then:

(1) w is periodic if and only if w′ is periodic.

(2) If w, w′ are periodic, and if γ, γ′ ⊂ HΛ are the geodesics in T associated to w, w′ respectively, then γ, γ′ are periodic and there is a singular solvisometry Φ: Q_{γ} → Q_{γ′} such that the following diagram coarsely commutes:

\xymatrix{P_{w} \ar[r]^{ϕ} \ar[d]_{F_{w}} & P_{w′} \ar[d]^{F_{w′}} \\ Q_{γ} \ar[r]^{Φ} & Q_{γ′}}

That is, d(F_{w′}(ϕ(x)), Φ(F_{w}(x))) ≤ A where A depends only on the quasiisometry constants of ϕ, not on w or w′.

We start by reducing the proposition to a statement about singular solvmanifolds by using the following result, which shows that various competing notions of “periodicity” are in fact equivalent:

Lemma 6.2 (Characterizing periodicity). Given a T_{H} geodesic w and the corresponding Teichmüller geodesic γ = γ_{w}, the following are equivalent:

(1) P_{w} is periodic, meaning that the group C_{w} is infinite.

(2) γ is periodic in T , meaning that the group Stab_{T} (γ) is infinite.

(3) Q_{γ} is periodic, meaning that the group C_{γ} = Isom(Q_{γ})/ Isom_{h}(Q_{γ}) is infinite.

Proof. We clearly have inclusions C_{w} ⊂ Stab_{T} (γ_{w}) ⊂ C_{γ} and so (1) implies (2) implies (3).

To prove (2) implies (1), suppose that γ is periodic in T , that is, γ is the axis of some pseudo-Anosov element ϕ ∈ M. Choose a point p ∈ γ, and consider the sequence ϕ^{n}(p), n > 0. Since H acts cocompactly on HΛ, there is a sequence ψ_{n} ∈ H such that {ψ_{n} ∘ ϕ^{n}(p) | n > 0} is a bounded subset of T . Since M acts properly on T , there exist m > n > 0 such that ψ_{n} ∘ ϕ^{n} = ψ_{m} ∘ ϕ^{m}. It follows that ϕ^{m−n} = ψ_{m}^{−1}ψ_{n} ∈ H, and so w is periodic in T_{H}.

To prove (3) implies (2), choose a horizontal leaf D_{γ} of Q_{γ} and so we have the split exact sequence

1 → Isom_{+}(D_{γ}) ≈ Isom_{h}(Q_{γ}) → Isom_{+}(Q_{γ}) → C_{γ+} → 1

where C_{γ+} is an infinite cyclic subgroup of index ≤ 2 in C_{γ}. We also have a finite index inclusion π_{1}(Σ) < Isom_{+}(D_{γ}). Since C_{γ+} acts by automorphisms of Isom_{+}(D_{γ}) it follows that C_{γ+} has a finite index subgroup stabilizing π_{1}(Σ); this subgroup is clearly identified with a subgroup of Stab_{T} (γ_{w}) and so the latter is infinite. ♢

Note from the proof that each of the inclusions C_{w} ⊂ Stab_{T} (γ_{w}) ⊂ C_{γ} has finite index; examples may be constructed in which any number of these inclusions is proper.

By combining Lemma 6.2 with Proposition 4.4, it follows immediately that Proposition 6.1 is reduced to the following:

Lemma 6.3. Given geodesics γ, γ′ in T , if γ is periodic, and if there exists a horizontal respecting quasi-isometry Q_{γ} → Q_{γ′}, then γ′ is periodic and there exists a singular solv isometry Q_{γ} → Q_{γ′}.

Remark. Our proof of Lemma 6.3 uses both Thurston’s hyperbolization theorem for mapping tori of pseudo-Anosov homeomorphisms [Ota96] as well as the geodesic pattern rigidity theorem of R. Schwartz [Sch97], and to apply these results we need to invoke periodicity of γ. Existence of a horizontal respecting quasi-isometry between Q_{γ} and Q_{γ′} ought to imply existence of a singular solvisometry, without assuming periodicity of γ; however, we do not know a proof.

6.1 Rigidity of periodic hyperplanes

The main step in the proof of Lemma 6.3 is Proposition 6.4 below, which shows that a periodic singular solv-manifold Q_{γ} has the rigidity property that its horizontal respecting quasi-isometry group equals its isometry group. Here is the basic setup.

For each periodic geodesic γ in T , fix a base horizontal leaf D_{γ} ⊂ Q_{γ}. Recall that π_{1}(Σ) acts as a deck transformation group on D_{γ}. The singular solv-metric on Q_{γ} induces an xy-structure on D_{γ} with respect to which we have an inclusion π_{1}(Σ) ↪ Isom_{+}(D_{γ}). There is a commutative diagram of short exact sequences

\xymatrix{1 \ar[r] & Isom_{+}(D_{γ}) \ar[r] & Isom(Q_{γ}) \ar[r] & C_{γ} \ar[r] & 1 \\ 1 \ar[r] & π_{1}(Σ) \ar[r] \ar@{^{(}->}[u] & Γ_{γ} \ar[r] \ar@{^{(}->}[u] & Stab_{T} (γ) \ar[r] \ar@{^{(}->}[u] & 1}

where each vertical arrow is a finite index inclusion. The group C_{γ} is either Z or D_{∞}. The quotient Q_{γ}/ Isom(Q_{γ}) is a closed 3-dimensional orbifold, equipped with an orbifold fibration over the circle (when C_{γ} = Z) or the interval orbifold (when C_{γ} = D_{∞}); in the latter case we choose D_{γ} so that it projects to a generic point of the interval orbifold, i.e. so that the stabilizer of D_{γ} in Isom(Q_{γ}) equals Isom_{+}(D_{γ}). It follows that the generic fiber of the quotient 3-orbifold Q_{γ}/ Isom(Q_{γ}) is the closed 2-orbifold O_{γ} = D_{γ}/ Isom_{+}(D_{γ}) equipped with the quotient xy-structure.

Note that the quotient 3-orbifold Q_{γ}/Γ_{γ} also fibers over the circle or the interval orbifold, with fiber Σ. There is a finite covering map Σ → O_{γ} and the fiber monodromy map on O_{γ} lifts to a pseudo-Anosov homeomorphism ϕ: Σ → Σ.

Let QI_{h}(Q_{γ}) be the subgroup of QI(Q_{γ}) represented by horizontal respecting quasi-isometries of Q_{γ}. Clearly there is a homomorphism

i: Isom(Q_{γ}) → QI_{h}(Q_{γ})

Proposition 6.4 (Rigidity of periodic hyperplanes). If γ is periodic then the homomorphism i: Isom(Q_{γ}) → QI_{h}(Q_{γ}) is an isomorphism.

Proof. A nontrivial isometry ϕ of Q_{γ} must map some vertical geodesic to a different vertical geodesic. Since any two vertical geodesics in Q_{γ} have infinite Hausdorff distance, ϕ is an infinite distance from the identity, so that i is injective (injectivity of i therefore is true regardless of periodicity).

We now prove that i is surjective: every horizontal respecting quasi-isometry of Q_{γ} is a bounded distance from an isometry.

Thurston’s geometrization theorem for pseudo-Anosov mapping tori gives a hyperbolic structure on the 3-dimensional orbifold Q_{γ}/ Isom(Q_{γ}). This yields a properly discontinuous, cocompact, isometric, faithful action

h: Isom(Q_{γ}) → Isom(H^{3})

and a quasi-isometric, h-equivariant homeomorphism

q: Q_{γ} → H^{3}

For each singular vertical geodesic ℓ ⊂ Q_{γ}, the image q(ℓ) is a quasigeodesic in H^{3}, which by the Morse-Mostow Lemma is a bounded Hausdorff distance (not depending on ℓ) from a unique geodesic in H^{3}; let L denote the set of all such geodesics in H^{3}. Let Isom(H^{3}, L) be the group of isometries of H^{3} which permute the collection L. Also, let F be the foliation of H^{3} obtained by pushing forward via q the horizontal foliation of Q_{γ}, and let Isom(H^{3}, L, F) be the subgroup of Isom(H^{3}, L) which coarsely respects F. Note that image(h) ⊂ Isom(H^{3}, L, F). Now we show that h factors through i.

For each horizontal respecting quasi-isometry ψ of Q_{γ}, we obtain a quasiisometry qψq^{−1} of H^{3}, inducing a homomorphism q̂: QI_{h}(Q_{γ}) → QI(H^{3}). Since ψ coarsely respects singular vertical geodesics and horizontal leaves in Q_{γ}, it follows that qψq^{−1} coarsely respects L and F.

Since qψq^{−1} coarsely respects L, since L is invariant under the cocompact isometry group image(h), and since there are only finitely many orbits of the action of image(h) on L, we may directly apply the main theorem of [Sch97] which says in this setting that qψq^{−1} is a bounded distance (not uniformly so) from a unique isometry of H^{3} which strictly respects L.

It follows that the image of q̂ is contained in Isom(H^{3}, L), and in fact in Isom(H^{3}, L, F). It’s evident that the composition

Isom(Q_{γ}) ⟶^{i} QI_{h}(Q_{γ}) ⟶^{q̂} Isom(H^{3}, L, F) ⊂ Isom(H^{3})

is identical with the homomorphism h. The homomorphism q̂ is obviously injective, and so to prove surjectivity of i it suffices to show that image(h) = Isom(H^{3}, L, F).

Consider the short exact sequence

1 → Isom_{+}(D_{γ}) → Isom(Q_{γ}) → C_{γ} → 1

Since Isom_{+}(D_{γ}) is normal in Isom(Q_{γ}), and since Isom(Q_{γ}) is identified via h with a finite index subgroup of Isom(H^{3}, L, F) (both being discrete and cocompact on H^{3}), it follows that Isom_{+}(D_{γ}) has a normalizer of finite index in Isom(H^{3}, L, F); choose coset representatives g_{1}, …, g_{n} of the normalizer. Each leaf of F is coarsely equivalent to Isom_{+}(D_{γ}), and g_{1}, …, g_{n} coarsely respects F, and so each conjugate subgroup

g_{1} Isom_{+}(D_{γ})g_{1}^{−1}, …, g_{n} Isom_{+}(D_{γ})g_{n}^{−1}

is coarsely equivalent to Isom_{+}(D_{γ}).

Now we apply an elementary lemma of [MSW00] which says that for a finite collection of subgroups in a finitely generated group, the coarse intersection of those subgroups is coarsely equivalent to their intersection. The intersection of the above conjugates of Isom_{+}(D_{γ}) is therefore coarsely equivalent to D_{γ}.

Another elementary lemma of [MSW00] says that in a finitely generated group, given subgroups A ⊂ B, if A, B are coarsely equivalent, then A has finite index in B. It follows that the intersection of the conjugates of Isom_{+}(D_{γ}) in Isom(H^{3}, L, F) has finite index in Isom_{+}(D_{γ}). Thus we obtain a normal subgroup N ⊂ Isom(H^{3}, L, F) of finite index in Isom_{+}(D_{γ}). The quotient group Isom(H^{3}, L, F)/N is a finite index supergroup of C_{γ} = Isom(Q_{γ})/ Isom_{+}(D_{γ}), and so the quotient is virtually cyclic. Since Isom(H^{3}, L, F) is a 3-orbifold group it follows that the quotient Isom(H^{3}, L, F)/N is either Z or D_{∞}.

The orbifold H^{3}/ Isom(H^{3}, L, F) therefore fibers over the circle or the interval orbifold, with generic fiber O, and with π_{1}(O) identified with N. Lifting this fibration to H^{3} we obtain an Isom(H^{3}, L, F) equivariant fibration coarsely equivalent to F. We may therefore replace F with this fibration, and so F is strictly invariant under Isom(H^{3}, L, F). The monodromy map on O is pseudo-Anosov. There is a finite index covering map Q_{γ}/ Isom(Q_{γ}) → H^{3}/ Isom(H^{3}, L, F), taking fibration to fibration. By uniqueness of pseudo-Anosov homeomorphisms in their isotopy classes [FLP^{+}79], it follows that the stable and unstable measured foliations for the monodromy map of O lift to the stable and unstable measured foliations for the monodromy map on the generic fiber O_{γ} of Q_{γ}/ Isom(Q_{γ}). But this shows that Isom(H^{3}, L, F) acts isometrically on Q_{γ}, proving that image(h) = Isom(H^{3}, L, F).

This proves Proposition 6.4. ♢

6.2 Proof of Lemma 6.3

Assume Q_{γ} is periodic and ϕ: Q_{γ} → Q_{γ′} is a horizontal respecting quasi-isometry. We’ll construct an isometry Φ: Q_{γ} → Q_{γ′}, the existence of which implies that Q_{γ′} is periodic.

Applying Proposition 5.2, we may move ϕ a bounded distance so that ϕ is a homeomorphism, respecting the horizontal foliations and the weak stable and unstable foliations. We may assume that the base horizontal leaves D_{γ} ⊂ Q_{γ}, D_{γ′} ⊂ Q_{γ′} are chosen so that ϕ(D_{γ}) = D_{γ′}.

Let A_{γ}: π_{1}(Σ) → Isom(Q_{γ}) and A_{γ′}: π_{1}(Σ) → Isom(Q_{γ′}) be the standard isometric actions, preserving each horizontal leaf. Let Q̂I_{h}(Q_{γ}) be the subsemigroup of Q̂I(Q_{γ}) that coarsely respects the horizontal foliation of Q_{γ}, and let

B_{γ} = ϕ^{−1} ∘ A_{γ′} ∘ ϕ: π_{1}(Σ) → Q̂I_{h}(Q_{γ})

be the conjugated action (its really an action, not just a quasi-action, because ϕ is a homeomorphism). Note that B_{γ} preserves each horizontal leaf of Q_{γ} as well as the strong stable and unstable foliations in that leaf; however, B_{γ} does not a priori preserve the invariant measures on those foliations.

Applying Proposition 6.4, we conclude that B_{γ} is a bounded distance from an isometric action, that is, there exists a homeomorphism ξ: Q_{γ} → Q_{γ} which moves each point a uniformly bounded distance, such that ξ^{−1} ∘ B_{γ} ∘ ξ is an isometric action of π_{1}(Σ) on Q_{γ}. The action ξ^{−1} ∘ B_{γ} ∘ ξ also preserves each horizontal leaf of Q_{γ} and the strong stable and unstable foliations in that leaf.

We claim that B_{γ} = ξ^{−1} ∘ B_{γ} ∘ ξ. Consider a horizontal leaf L of Q_{γ} and a point x ∈ L. Let ℓ^{s}, ℓ^{u} be the strong stable and unstable leaves in L passing through x, and so x = ℓ^{s} ∩ ℓ^{u}. Consider an element β ∈ B_{γ}(π_{1}(Σ)). We know that β(L) = ξ^{−1}βξ(L) = L. We also know that β(ℓ^{s}) and ξ^{−1}βξ(ℓ^{s}) are both strong stable leaves in L, and they are a bounded distance from each other, implying that β(ℓ^{s}) = ξ^{−1}βξ(ℓ^{s}). Similarly, β(ℓ^{u}) = ξ^{−1}βξ(ℓ^{u}). Therefore, β(x) = ξ^{−1}βξ(x), proving the claim.

It follows from the claim that B_{γ} does, in fact, preserve the invariant measures on the strong stable and unstable foliations in each horizontal leaf of Q_{γ}.

Fix a horizontal leaf L ⊂ Q_{γ} and let L′ = ϕ(L) ⊂ Q_{γ′}. So ϕ takes the strong stable and unstable foliations in L to those in L′. Let f^{s}, f^{u} be the strong stable and unstable measured foliations in L, and let f′^{s}, f′^{u} the strong stable and unstable measured foliations in L′. The map ϕ pushes the transverse measures on f^{s}, f^{u} forward to new transverse measures on f′^{s}, f′^{u}.

Since B_{γ} = ϕ^{−1}A_{γ′}ϕ acts isometrically on the transverse measures of f^{s}, f^{u}, it follows that A_{γ′} acts isometrically on both the old and the new transverse measures on f′^{s}, f′^{u}. However, f′^{s}, f′^{u} are uniquely ergodic with respect to the A_{γ′} action, i.e. they have projectively unique transverse measures invariant under the A_{γ′} action. This follows from the fact that the Teichmüller geodesic γ′ is cobounded, together with a theorem of H. Masur that if ξ ∈ PMF is not uniquely ergodic then any Teichmüller ray with ending foliation ξ is not cobounded [Mas80]. Hence the old and new transverse measures on f′^{s}, f′^{u} differ by multiplicative constants. These multiplicative constants are inverses to each other, because the action A_{γ′} has cofinite area. It follows that ϕ restricts to an xy-affine isomorphism from D_{γ} to D_{γ′}. This implies in turn that the quasi-isometry ϕ: Q_{γ} → Q_{γ′} may be altered a bounded amount, moving each point a fixed amount up or down in its vertical geodesic, to obtain an isometry Φ: Q_{γ} → Q_{γ′}.

It follows that D_{γ′} has an extra, nonisometric, affine symmetry, and so γ′ is periodic.

This completes the proof of Lemma 6.3 and therefore also of Proposition 6.1.

7 The endgame: computing QI(Γ_{H}) and Comm(Γ_{H})

7.1 An injection QI(X_{H}) → QSym(S^{1})

Let QSym(S^{1}) denote the group of quasisymmetric homeomorphisms of the circle S^{1}. It is well-known that the extension of a quasi-isometry of H^{2} to S^{1} induces an isomorphism QI(H^{2}) ≈ QSym(S^{1}): boundary extension of quasi-isometries defines an injection from the quasi-isometry group of any word hyperbolic group to the homeomorphism group of its boundary; and quasi-symmetric homeomorphisms of S^{1} are exactly the extensions of quasi-isometries of H^{2}. The key to all of this is the theorem of Ahlfors and Beurling [AB56] that quasisymmetric homeomorphisms of S^{1} are exactly the extensions of quasiconformal homeomorphisms of the unit disc.

Fix once and for all a base fiber D_{0} = π^{−1}(τ_{0}) of X_{H}, τ_{0} ∈ T_{H}. Identify D_{0} with H^{2}, so ∂D_{0} is identified with S^{1}. Then there is a map Ψ: QI(X_{H}) → QSym(S^{1}), defined as follows. Given a quasi-isometry f: X_{H} → X_{H}, the image f(D_{0}) is Hausdorff close to some fiber D′. Consider the composition

D_{0} → f(D_{0}) → D′ → D_{0}

where the first map is f and the other maps are closest point projections. Then the composition D_{0} → D_{0} is clearly a quasi-isometry, and so induces an element QSym(S^{1}). Hence we have a map Ψ: QI(X_{H}) → QSym(S^{1}), which is easily seen to be well-defined, and in fact a homomorphism.

Proposition 7.1. The homomorphism Ψ: QI(X_{H}) → QSym(S^{1}) is injective.

Proof. By Propositions 4.1 and 4.2, and the fact that the projection X_{H} → T_{H} induces an isometry between the space of horizontal leaves of X_{H} with the Hausdorff metric and the quotient tree T_{H}, it follows that any quasi-isometry f: X_{H} → X_{H} induces a quasi-isometry of T_{H} and so f induces a homeomorphism Ξ(f): Λ → Λ of the Cantor set Λ = ∂T_{H} ⊂ PMF.

Suppose that Ψ(f) ∈ QSym(S^{1}) is the identity map on S^{1}.

We claim that the induced map Ξ(f): Λ → Λ is the identity. It follows that the induced quasi-isometry T_{H} → T_{H} is a bounded distance from the identity, and so f takes each horizontal leaf of X_{H} a bounded distance from itself; and since the induced boundary map of that leaf is the identity, it follows that f takes each point a bounded distance from itself, proving the proposition.

For proving the claim (and for later purposes) we review the well-known embedding of MF into the space of π_{1}(Σ)-invariant measures on the “Möbius band beyond infinity” of the hyperbolic plane. That is, consider the double set of the circle, DS^{1} = {{x, y} ⊂ S^{1} | x ≠ y}; with respect to the Klein model D_{0} = H^{2} ⊂ RP^{2}, the usual duality gives a bijection between DS^{1} and the Möbius band beyond infinity RP^{2} − \overline{H}^{2}.

Let M(DS^{1}) be the space of Borel measures on DS^{1} with the weak^{∗} topology, and let PM(DS^{1}) be the space of projective classes of elements of M(DS^{1}). The space MF embeds into the space of π_{1}(Σ)-invariant elements of M(DS^{1}), by lifting a measured foliation on Σ to a π_{1}(Σ)-invariant measured foliation on D_{0}, and then identifying each leaf of the lifted foliation with the correspond pair of endpoints in DS^{1}. The space PMF therefore embeds as π_{1}(Σ)-invariant elements of PM(DS^{1}). Given b ∈ PMF let μ_{b} ∈ PM(DS^{1}) be the corresponding projective class of measures, let supp(μ_{b}) ⊂ DS^{1} be the support, and let E_{b} = |supp(μ_{b})| be the union of all the pairs in supp(μ_{b}). Note that E_{b} may also be described as the endpoint set of b, the set of endpoints in S^{1} of the leaves of the measured foliation F̃_{b} on D_{0} obtained by lifting any measured foliation F_{b} on Σ that represents b.

By Proposition 5.2, for all b ∈ Λ and all quasi-isometries ϕ of X_{H} we have

Ψ(ϕ)(E_{b}) = E_{Ξ(ϕ)(b)}

From our assumption that Ψ(f) is the identity on S^{1} it follows that Ψ(f)(E_{b}) = E_{b}, for all b ∈ Λ. Note however that if b ≠ b′ ∈ Λ then E_{b} ∩ E_{b′} = ∅, because the projective classes of F_{b}, F_{b′} in PMF are connected by a Teichmüller geodesic in HΛ, and so the measured foliations F_{b}, F_{b′} can be chosen to be transverse in Σ; the lifted foliations F̃_{b}, F̃_{b′} are therefore transverse in D_{0} and so their endpoint sets E_{b}, E_{b′} in S^{1} are disjoint. It follows that Ξ(f)(b) = b for all b ∈ Λ. ♢

7.2 The orbifold O_{H} associated to a Schottky group H

Our goal in this subsection and the next is to compute the quasi-isometry group QI(Γ_{H}). Using the injection Ψ: QI(Γ_{H}) → QSym(S^{1}) provided by Proposition 7.1, our computation will consist of an explicit description of the subgroup Ψ(QI(Γ_{H})) < QSym(S^{1}); without further mention we shall identify QI(Γ_{H}) with this subgroup.

The first step in the computation of QI(Γ_{H}) is to find a natural orbifold subcover Σ → O_{H} associated to a Schottky subgroup H < M(Σ). The orbifold O_{H} is an important invariant of H; it is the smallest subcover of Σ to which H descends as a subgroup of M(O_{H}).

Recall we have fixed a base fiber D_{0} of X_{H} over a base point τ_{0} ∈ T_{H}, and we identify D_{0} with the universal cover Σ̃.

To each periodic hyperplane P_{w} we associate a finite-index supergroup of π_{1}(Σ) in QSym(S^{1}), as follows. In the associated singular solv-manifold Q_{γ_{w}} pick any horizontal leaf D_{w}, which has an induced xy structure with xy-affine automorphism group denoted Aff(D_{w}). Note that Aff(D_{w}) acts by quasi-isometries of the xy-metric on D_{w}. There is a canonical quasi-isometry from D_{w} to D_{0}: restrict the canonical horizontal respecting quasi-isometry Q_{γ_{w}} → P_{w} to the horizontal leaf D_{w}, giving a map D_{w} → X_{H}, and then take a closest point map to D_{0}. In this way we obtain an inclusion

Aff(D_{w}) ⊂ QI(D_{0}) = QSym(S^{1})

Note that the image of this inclusion is independent of the choice of horizontal leaf D_{w} in Q_{γ_{w}}, because if we chose another horizontal leaf D′_{w} then the vertical flow on Q_{γ_{w}} induces an xy-affine homeomorphism D′_{w} → D_{w} in the correct quasi-isometry class.

Recall that we have a short exact sequence

1 → Isom_{+}(D_{w}) → Aff(D_{w}) → C_{γ_{w}} → 1

with C_{γ_{w}} ≡ Z or D_{∞}. Under the injection Aff(D_{w}) ↪ QSym(S^{1}), the group Isom_{+}(D_{w}) is a finite index supergroup of π_{1}(Σ). We have a quotient orbifold O_{w} = D_{w}/ Isom_{+}(D_{w}) with fundamental group π_{1}(O_{w}) ≈ Isom_{+}(D_{w}), and associated to the inclusion π_{1}(O_{w}) ⊂ π_{1}(Σ) there is a finite orbifold covering map Σ → O_{w}.

In the group QSym(S^{1}), take the infinite intersection of the groups Isom_{+}(D_{w}) over all periodic lines w in the tree T_{H}, and note that this group must be a finite index supergroup of π_{1}(Σ), in fact it is the fundamental group π_{1}(O_{H}) of an orbifold O_{H} which Σ finitely covers, the smallest orbifold covered by Σ which in turn covers each orbifold O_{w}.

π_{1}(O_{H}) = ⋂_{w⊂T_{H} ; w periodic} Isom_{+}(D_{w})

Note that O_{H} is the smallest subcover of Σ such that the Schottky group H ⊂ M(Σ) descends via the covering map Σ → O_{H} to a free subgroup of M(O_{H}). To be precise, since H is free we may choose a section σ: H → M(Σ, p) ⊂ QSym(S^{1}), and the image group σH acting by conjugation on subgroups of QSym(S^{1}) permutes the collection of subgroups Isom_{+}(D_{w}) and so σH ⊂ Aut(π_{1}(O_{H})) = M(O_{H}, p). Projecting to M(O_{H}) we obtain the free subgroup H′, and a section σ′: H′ → M(O_{H}, p) < QSym(S^{1}) so that σH = σ′H′.

Remark. Orbifold mapping class groups obey many of the properties of surface mapping class groups. The group M(O_{H}) is defined as the group of orbifold homeomorphisms of O_{H} modulo those which are isotopic to the identity through orbifold homeomorphisms. Just as with mapping class groups of surfaces, choosing a generic point p ∈ O_{H} we obtain an isomorphism of short exact sequences

[figure: 1; π_{1}(O_{H}); M(O_{H}, p); π; M(O_{H}); 1; 1; π_{1}(O_{H}); Aut(π_{1}(O_{H})); π; Out(π_{1}(O_{H})); 1]

The proof that this isomorphism exists follows the same lines as the proof for surfaces, using the fact that π_{1}(O_{H}) is centerless.

By choosing a lift of p to the universal cover of D_{0} = Õ_{H} and lifting each element of M(O_{H}, p), we identify M(O_{H}, p) with a subgroup of QI(D_{0}) = QSym(S^{1}).

In fact H′ < M(O_{H}) is a Schottky subgroup. To see why, consider the limit set Λ(H) ⊂ PMF (Σ). Moving into S^{1}, the corresponding subset {μ_{b} | b ∈ Λ(H)} of PM(DS^{1}) is invariant under σH. Notice that for each b ∈ Λ(H), the element μ_{b} ∈ PM(S^{1}), which is invariant under π_{1}(Σ), is also invariant under the larger group π_{1}(O_{H}). When b is an endpoint of a periodic geodesic w in T_{H} this is obvious, because μ_{b} is invariant under π_{1}(O_{w}) ⊃ π_{1}(O_{H}). But periodic endpoints are dense in Λ(H), and so each μ_{b}, b ∈ Λ(H) is invariant under π_{1}(O_{H}). We therefore obtain a continuous embedding of Λ(H) in PMF (O_{H}), whose image we denote Λ(H′); this embedding has the property that

{μ_{b} | b ∈ Λ(H)} = {μ_{b′} | b′ ∈ Λ(H′)} in PM(DS^{1})

Moreover, since σ′H′ = σH, it follows that σ′H′ permutes the elements of the above set, and so Λ(H′) is in fact invariant under the action of H′. For any pair ξ ≠ η ∈ Λ(H) and corresponding pair ξ′, η′ ∈ Λ(H′), each xy-structure on Σ corresponding to a point along the Teichmüller geodesic \overleftrightarrow{(ξ, η)} ⊂ T (Σ) lifts to an xy-structures on D_{0} which is invariant under π_{1}(O_{H}), and we therefore obtain a Teichmüller geodesic \overleftrightarrow{(ξ′, η′)} in T (O_{H}); the π_{1}(O_{H})-invariance follows by an argument similar to the one just above where we showed invariance for elements μ_{b}, b ∈ Λ(H). The union of these geodesics over all pairs ξ′ ≠ η′ ∈ Λ(H′) is denoted HΛ(H′) as usual. From this construction we see that the action of H′ on Λ(H′) ∪ HΛ(H′) agrees with the action of H on Λ(H) ∪ HΛ(H). In particular, all the requirements in Theorem 3.1 for H′ to be convex cocompact with limit set Λ(H′) are satisfied, and so H′ is a Schottky subgroup of M(O_{H}).

Remark. The fact that the orbifold O_{H} supports a pseudo-Anosov mapping class puts some restrictions on its topology: the underlying surface of O_{H} must have empty boundary, because otherwise the collection of peripheral curves would be invariant under any mapping class, violating the existence of pseudo-Anosov mapping classes. It follows that O_{H} is a closed surface with cone singularities.

7.3 Computing the quasi-isometry group

In this section we compute QI(Γ_{H}). Let C denote the relative commensurator of H′ in M(O_{H}):

C = Comm_{M(O_{H})}(H′)

Form the extension group Γ_{C}, a subgroup of M(O_{H}, p) as the following diagram shows:

[figure: 1; π_{1}(O_{H}); Γ_{C}; C; 1; 1; π_{1}(O_{H}); M(O_{H}, p); π; M(O_{H}); 1]

The group Γ_{C} may therefore be regarded as a subgroup of QSym(S^{1}). Here is our computation of QI(Γ_{H}):

Theorem 7.2. In QSym(S^{1}) we have

QI(Γ_{H}) = Γ_{C}.

Moreover, the subgroup π_{1}(O_{H}) consists of those classes of quasi-isometries of X_{H} which coarsely preserve each horizontal leaf of X_{H}.

Proof. To justify this computation, first we show QI(Γ_{H}) ⊂ Γ_{C}. Consider a quasiisometry f: X_{H} → X_{H}, regarded as an element of QSym(S^{1}); we must show that f ∈ Γ_{C}.

By Lemma 4.3 the quasi-isometry f coarsely respects the horizontal foliation of X_{H} and so f lies over a quasi-isometry of T_{H}, also denoted f. From Proposition 6.1 it follows that f induces a permutation on the set of periodic hyperplanes of X: given a periodic geodesic w in T_{H}, if we let w′ be the geodesic in T_{H} coarsely equivalent to f(w), then f takes P_{w} to P_{w′} and Q_{γ_{w}} to Q_{γ_{w′}}. It follows that the conjugation action of f on subgroups of QSym(S^{1}) takes Isom(Q_{γ_{w}}) = Aff(D_{w}) to Isom(Q_{γ_{w′}}) = Aff(D_{w′}). Any isometry of Q_{γ_{w}} which preserves each leaf of the horizontal foliation is conjugated by f to a similar isometry of Q_{γ_{w′}}, and so conjugation by f takes Isom_{h}(Q_{γ_{w}}) = Isom_{+}(D_{w}) to Isom_{h}(Q_{γ_{w′}}) = Isom_{+}(D_{w′}). In other words, conjugation by f preserves the collection of subgroups

{Isom_{+}(D_{w}) | w ⊂ T_{H} is periodic}

Intersecting this collection it follows that conjugation by f in the group QSym(S^{1}) preserves the subgroup π_{1}(O_{H}), acting as an automorphism of that subgroup. In other words, in QSym(S^{1}) we have f ∈ Aut(π_{1}(O_{H})) = M(O_{H}, p).

Now we show that the image πf ∈ M(O_{H}) lies in C = Comm_{M(O_{H})}(H′). We need the following fact:

Theorem 7.3 (Commensurators of Schottky groups). Let O be a closed orbifold and H′ a Schottky subgroup of M(O). Then the relative commensurator Comm_{M(O)}(H′) of H′ in M(O) is equal to the subgroup of M(O) stabilizing the limit set Λ(H′), and H′ has finite index in Comm_{M(O)}(H′).

Proof. Let I be the subgroup of M(O) stabilizing Λ(H′), and so H′ < I. Let HΛ(H′) be the weak convex hull of Λ(H′) (see Theorem 3.1). Clearly I is also the subgroup of M(O) stabilizing HΛ(H′). By Theorem 3.1 the group H′ acts cocompactly on HΛ(H′), and so I acts cocompactly on HΛ(H′). Clearly I acts properly on HΛ(H′). It follows that H′ has finite index in I, which immediately implies I < Comm_{M(O)}(H′).

For the opposite inclusion, suppose Φ ∈ M(O) − I, and so Φ(Λ(H′)) ≠ Λ(H′). The limit set of the Schottky subgroup ΦH′Φ^{−1} is Φ(Λ(H′)). Since Λ(H′) is closed, Φ(Λ(H′)) − Λ(H′) is open in Φ(Λ(H′)). Since fixed points of pseudo-Anosov elements of ΦH′Φ^{−1} are dense in Φ(Λ(H′)), there exists a pseudo-Anosov element Ψ ∈ ΦH′Φ^{−1} having a fixed point not in Λ(H′). Infinitely many powers of Ψ are therefore not in H′, and so H′ ∩ ΦH′Φ^{−1} has infinite index in ΦH′Φ^{−1}. ♢

We showed above that H′ is a Schottky subgroup of M(O_{H}) and so by Theorem 7.3 it remains to show that πf ∈ M(O_{H}) acting on PMF (O_{H}) leaves the set Λ(H′) invariant. For this purpose it suffices to show that the action of f ∈ QSym(S^{1}) leaves the set {μ_{b′} | b′ ∈ Λ(H′)} invariant. However, we know that this set equals {μ_{b} | b ∈ Λ(H)}, and we also know that f leaves this set invariant: f permutes the elements of this set which are endpoints of periodic geodesics in T_{H}, by Proposition 6.1, but the endpoints of periodic geodesics are dense.

This completes the proof of the inclusion QI(Γ_{H}) ⊂ Γ_{C}.

For the opposite inclusion, consider the following commutative diagram of short exact sequences:

\xymatrix{1 \ar[r] & π_{1}(Σ) \ar[r] \ar@{^{(}->}[d] & Γ_{H} \ar[r] \ar[d] & H \ar[r] \ar@{^{(}->}[d] & 1 \\ 1 \ar[r] & π_{1}(O_{H}) \ar[r] & Γ_{C} \ar[r] & C \ar[r] & 1}  (7.1)

The left vertical arrow is a finite index injection because Σ → O_{H} is a finite covering. The right vertical arrow is a finite index injection by Theorem 7.3 using the isomorphism H ≈ H′. It follows that the middle vertical arrow is a finite index injection. But this shows that Γ_{C} is a finite index supergroup of Γ_{H} in QSym(S^{1}), and so Γ_{C} ⊂ QI(Γ_{H}).

This completes the computation of QI(Γ_{H}).

To complete the proof of Theorem 7.2, it remains to prove that an element of QI(Γ_{H}) = Γ_{C} is in the normal subgroup π_{1}(O_{H}) if and only if it coarsely preserves each horizontal leaf of X_{H}. Evidently π_{1}(O_{H}) coarsely preserves each horizontal leaf. Suppose conversely that the quasi-isometry f: X_{H} → X_{H} coarsely preserves each horizontal leaf; let [f] ∈ QI(Γ_{H}) be the coarse equivalence class of f. It follows that f coarsely respects each hyperplane of X_{H}, in particular each periodic hyperplane P_{w}, and so we have [f] ∈ Aff(D_{w}) for each periodic line w in T_{H}. But we can say more, as follows. The inclusion map Γ_{H} → QI(Γ_{H}), being injective and with finite index image, is a quasi-isometry, and we therefore obtain a quasi-isometry QI(Γ_{H}) → X_{H}. Using this quasi-isometry we may conjugate the left action of QI(Γ_{H}) on itself to obtain a quasi-action of QI(Γ_{H}) on X_{H}. In particular, we obtain a sequence of uniform quasi-isometries f_{n}: X_{H} → X_{H}, with [f_{n}] = [f]^{n}. Each of the f_{n} coarsely preserves each horizontal leaf of X_{H}, and note that the coarseness constant is uniform independent of n and of the leaf, by application of Proposition 4.1 using uniformity of the quasi-isometry constants of f_{n}. It follows that each f_{n} coarsely preserves each periodic hyperplane P_{w}, and f_{n} coarsely preserves each horizontal leaf of P_{w}, again with uniform coarseness constants independent of n. This implies that [f] ∈ Isom_{+}(D_{w}). We therefore have

[f] ∈ ⋂_{w} Isom_{+}(D_{w}) = π_{1}(O_{H})

♢

7.4 Computing the commensurator group

In this section we use our knowledge of quasi-isometries of Γ_{H} to compute the commensurator of Γ_{H} and complete the proof of Theorem 1.3. This is the first instance we know of where this technique of computing a commensurator group is used. We prove, in Theorem 7.4, that when the left multiplication homomorphism Γ → QI(Γ) is an injection with finite index image, the natural homomorphism i: Comm(Γ) → QI(Γ) is an isomorphism. Combining with Theorem 7.2 we obtain the desired computation of Comm(Γ_{H}). We do not have a proof of the computation of Comm(Γ_{H}) without going through all of the work required to understand quasiisometries; a purely algebraic computation of Comm(Γ_{H}) would be interesting.

For any finitely-generated group Γ, a commensuration ϕ: G_{1} → G_{2} extends to a quasi-isometry of Γ by precomposing ϕ with a closest point projection Γ → G_{1} and postcomposing with inclusion G_{2} → Γ. The coarse equivalence class of this extension is well-defined, giving a natural homomorphism i: Comm(Γ) → QI(Γ). Note that we have a commutative triangle

\xymatrix{Γ \ar[r]^{C: g↦\{C_{g}\}} \ar[rr]^{L: g↦[L_{g}]} & Comm(Γ) \ar[r]^{i} & QI(Γ)}

where L_{g} is the left multiplication map x → gx with coarse equivalence class [L_{g}] ∈ QI(Γ), and C_{g} is the conjugation automorphism x → gxg^{−1} with equivalence class {C_{g}} ∈ Comm(Γ); commutativity follows because right multiplication by g^{−1} moves each point in Γ a bounded amount.

Theorem 7.4. Given a finitely generated group Γ, if L: Γ → QI(Γ) is an injection with finite index image, then i: Comm(Γ) → QI(Γ) is an isomorphism.

This theorem, proved in collaboration Kevin Whyte, is broken into three steps:

(1) injectivity of i: Comm(Γ) → QI(Γ);

(2) construction of an injective map Ψ: QI(Γ) → Comm(Γ);

(3) the proof that Ψ ∘ i is the identity on Comm(Γ).

Step 1: Injectivity of i.

Proposition 7.5 (K. Whyte). If Γ is any finitely generated group then the natural map i: Comm(Γ) → QI(Γ) is injective.

Proof. Suppose ϕ: G_{1} → G_{2} is a commensuration of Γ such that i(ϕ) equals the identity in QI(Γ). Then there is a bounded function δ: Γ → Γ so that for all g ∈ G_{1} we have

ϕ(g) = g δ(g)

Since δ is bounded, the cardinality M = # image(δ) is finite.

Plugging the above equation into ϕ(gh) = ϕ(g)ϕ(h) gives

h^{−1}δ(g)h = δ(gh)δ(h)^{−1}

Note that this is true for all h ∈ G_{1}, and the right hand side takes on at most M^{2} values. This implies that the centralizer of δ(g) in G_{1} has index at most M^{2}. Intersecting all subgroups of G_{1} of index ≤ M^{2} gives a finite index subgroup H < G_{1} which commutes with each δ(g). From the above equation it follows that δ is a homomorphism on H. Since image(δ) is finite it follows that δ has finite index kernel ker(δ) in H, and so ker(δ) has finite index in G_{1} and in Γ. In other words, ϕ(g) = g on the finite index subgroup ker(δ) of Γ, and so ϕ represents the identity element of Comm(Γ). ♢

For any finitely generated group Γ the kernel of C: Γ → Comm(Γ) is the virtual center VZ(Γ) consisting of all g ∈ Γ whose centralizer has finite index in Γ. Together with Proposition 7.5 it follows that injectivity of L: Γ → QI(Γ) is equivalent to the triviality of VZ(Γ).

Step 2: An injection Ψ: QI(Γ) → Comm(Γ). Identifying Γ with its image L(Γ) < QI(Γ), a finite index subgroup, it follows that any automorphism of QI(Γ) restricts to a commensuration of Γ. In particular, given F ∈ QI(Γ), the inner automorphism G ↦ F GF^{−1} of QI(Γ) restricts to a commensuration Ψ_{F} of Γ, giving a well-defined homomorphism Ψ: QI(Γ) → Comm(Γ). Keeping in mind the definition of L: Γ → QI(Γ), this means that for each F ∈ QI(Γ) and each x ∈ Γ we have the following equation in QI(Γ):

F · [L_{x}] · F^{−1} = [L_{Ψ_{F} (x)}]

To prove that Ψ is an injection, given F = [f] ∈ QI(Γ) suppose that Ψ_{F} is the identity map when restricted to a finite index subgroup G_{1} of Γ. Then for all x ∈ G_{1} we have

F · [L_{x}] · F^{−1} = [L_{x}]
F · [L_{x}] = [L_{x}] · F
[f ∘ L_{x}] = [L_{x} ∘ f]

which means that there exists a bounded function δ: Γ → Γ such that

f(xy) = xf(y)δ(y), x, y ∈ G_{1}

Plugging in y = 1 we get

f(x) = xf(1)δ(1), x ∈ G_{1}

This shows that the function f is a bounded distance from the identity map on G_{1}, and so F is the identity element in QI(Γ).

Step 3: Ψ∘i: Comm(Γ) → Comm(Γ) is the identity. Given a commensuration ϕ of Γ, we obtain a commensuration ϕ′ = Ψ_{i(ϕ)} satisfying the formulas

[ϕ ∘ L_{x} ∘ ϕ^{−1}] = [L_{ϕ′(x)}]
[ϕ ∘ L_{x} ∘ ϕ^{−1} ∘ L_{ϕ′(x^{−1})}] = [Id]

for all x in a certain finite index subgroup of Γ. We must prove that ϕ and ϕ′ agree on a further finite index subgroup of Γ.

From the above equation it follows that there is a bounded function δ: Γ → Γ such that for all h in a certain finite index subgroup H of Γ we have:

ϕ(x · ϕ^{−1}(ϕ′(x^{−1}) · h) = h · δ(h)
ϕ(x) · ϕ′(x^{−1}) · h = h · δ(h)
h^{−1} · (ϕ(x) · ϕ′(x^{−1})) · h = δ(h)

This shows that ϕ(x) · ϕ′(x^{−1}) has only finitely many conjugates by elements of H, and so the centralizer of ϕ(x) · ϕ′(x^{−1}) in H has finite index in H. It follows that the centralizer of ϕ(x) · ϕ′(x^{−1}) in Γ has finite index in Γ, which by definition of VZ(Γ) gives ϕ(x) · ϕ′(x^{−1}) ∈ VZ(Γ). But since L: Γ → QI(Γ) is injective, the virtual center VZ(Γ) is trivial, showing that ϕ(x) = ϕ′(x) for all x in a finite index subgroup of Γ.

Combining steps 1–3 it follows that Ψ: QI(Γ) → Comm(Γ) is surjective, and so Ψ is an isomorphism with inverse i: Comm(Γ) → QI(Γ), finishing the proof of Theorem 7.4.

Remark. The proof of Theorem 7.4 yields a more general conclusion: if Γ is a finitely generated group whose virtual center is trivial, then Comm(Γ) is isomorphic to the relative commensurator of L(Γ) in QI(Γ). The condition that the virtual center be trivial cannot be dropped: for example, a finite group has trivial quasi-isometry group but rarely is its abstract commensurator group trivial.

7.5 Proving Theorem 1.1 and Theorem 1.2

Theorem 1.2 follows immediately from Theorem 1.3 by a standard technique (see, e.g. [Sch96]). The basic observation we need says that if Γ is a finitely generated group and if the homomorphism Γ → QI(Γ) has finite cokernel and kernel, then for any finitely generated group H and any quasi-isometry ϕ: H → Γ the induced homomorphism ϕ_{∗}: H → QI(Γ) has finite index kernel and cokernel. As a consequence, the groups H and Γ are weakly commensurable, because their images are commensurable in QI(Γ).

We now prove Theorem 1.1. Let H_{i} < M(Σ_{g_{i}}), i = 1, 2, be Schottky groups of rank ≥ 2 with g_{i} ≥ 2. We must prove the equivalence of the following four statements:

(1) Γ_{H_{1}} and Γ_{H_{2}} are quasi-isometric.

(2) Γ_{H_{1}} and Γ_{H_{2}} are abstractly commensurable.

(3) There is an isomorphism O_{H_{1}} ≈ O_{H_{2}} such that in the group M(O_{H_{1}}) = M(O_{H_{2}}) the Schottky subgroups H_{1} and H_{2} are commensurable, meaning that H_{1} ∩ H_{2} has finite index in each of H_{1} and H_{2}.

(4) There is an isomorphism O_{1} ≈ O_{2} such that in the group M(O_{1}) = M(O_{2}) the Schottky groups H_{1} and H_{2} have the same limit set in the Thurston boundary of the Teichmüller space T (O_{1}) = T (O_{2}).

We also add in a fifth equivalent statement:

(5) The groups QI(Γ_{H_{1}}) and QI(Γ_{H_{2}}) are isomorphic.

The equivalence of statements (1) and (2) and (5) follows immediately from Theorem 1.3 using the fact that a quasi-isometry between two groups induces an isomorphism between their quasi-isometry groups.

The fact that (3) implies (2) is an immediate consequence of the commutative diagram 7.1 applied to Γ_{H_{1}} and to Γ_{H_{2}}.

To prove that (1) implies (3), suppose that there is a quasi-isometry Γ_{H_{1}} → Γ_{H_{2}}, which induces an isomorphism of quasi-isometry groups QI(Γ_{H_{1}}) ≈ QI(Γ_{H_{2}}). Consider, for each i = 1, 2, the short exact sequence

1 → π_{1}(O_{i}) → QI(Γ_{H_{i}}) → C_{i} → 1

where as usual O_{i} = O_{H_{i}} is the smallest orbifold subcover of Σ_{g_{i}} to which H_{i} descends, the subgroup C_{i} < M(O_{i}) is the relative commensurator of H_{i} in M(O_{i}), and H_{i} has finite index in C_{i}.

We claim that the isomorphism between QI(Γ_{H_{1}}) and QI(Γ_{H_{2}}) must take π_{1}(O_{1}) to π_{1}(O_{2}). This provides an isomorphism O_{1} ≈ O_{2} such that the induced isomorphism M(O_{1}) ≈ M(O_{2}) takes C_{1} to C_{2}, and statement (3) immediately follows.

To prove the claim, consider the model space X_{H_{i}} with its horizontal foliation. By Theorem 7.2 the subgroup π_{1}(O_{i}) of QI(Γ_{H_{i}}) ≈ QI(X_{H_{i}}) consists of quasiisometries which coarsely preserve each leaf of the horizontal foliation. The quasiisometry X_{H_{1}} → X_{H_{2}} coarsely respects horizontal foliations by Lemma 4.3, and so the coarse leaf preserving elements of QI(X_{H_{1}}) are taken bijectively by the isomorphism QI(X_{H_{1}}) ↔ QI(X_{H_{2}}) to the coarse leaf respecting quasi-isometries of X_{H_{2}}. In other words, π_{1}(O_{1}) is taken to π_{1}(O_{2}).

Finally, the equivalence of (3) and (4) follows immediately from Theorem 7.3, completing the proof of Theorem 1.1. ♢

8 Closing remarks

8.1 Surfaces versus orbifolds

As remarked in the introduction, the universe of groups quasi-isometric to (orbifold)- by-(virtual Schottky) groups is exactly the same as the universe of groups quasiisometric to (surface)-by-(Schottky) groups:

Proposition 8.1. Given a closed orbifold O and a virtual Schottky subgroup N < M(O), consider the extension Γ_{N} defined by

1 → π_{1}(O) → Γ_{N} → N → 1

There exists a closed oriented surface Σ of genus g ≥ 2, and a Schottky subgroup H < M(Σ), such that the group Γ_{H} = π_{1}(Σ) ⋊ H has finite index in Γ_{N}, and so the two groups are quasi-isometric.

Proof. There is a Schottky subgroup H′ < N of finite index; it follows that Γ_{H′} has finite index in Γ_{N}. Choose a splitting H′ → M(O, p), consider the action of H′ on π_{1}(O) by automorphisms. Choose a finite surface cover Σ′ with corresponding subgroup π_{1}(Σ′) < π_{1}(O), and consider the orbit of π_{1}(Σ′) under the action of H′. This orbit consists of a finite collection of finite index subgroups of π_{1}(O), whose intersection is a finite index subgroup corresponding to a surface group Σ which is a cover of Σ′. The group H′ < M(O, p) lifts to a subgroup H < M(Σ, p), which projects to a Schottky subgroup of M(Σ). The group Γ_{H} = π_{1}(Σ) ⋊ H is therefore a (surface)-by-(Schottky) group with finite index in Γ_{H′}, and so also in the original group Γ_{N}. ♢

8.2 Fibered hyperbolic 3-manifold groups

As mentioned in the footnote on page 3, the method of proof of Theorem 1.1 shows that if two word hyperbolic surface-by-free groups are quasi-isometric then they are horizontal respecting quasi-isometric, indeed they are horizontal respecting commensurable, as long as the free group has rank ≥ 2. When the free group is infinite cyclic this fails: all hyperbolic 3-manifolds fibering over the circle have quasi-isometric fundamental groups; but there exist fibered hyperbolic 3-manifolds which are not abstract commensurable—take, for example, an arithmetic example and a non-arithmetic example.

Nevertheless, we do obtain a classification of fundamental groups of fibered hyperbolic 3-manifold groups up to horizontal respecting quasi-isometry. Namely, let Σ be a closed surface of genus g ≥ 2, let ψ ∈ M(Σ) be a pseudo-Anosov mapping class generating an infinite cyclic subgroup ⟨ψ⟩ of M(Σ), and let Γ_{⟨ψ⟩} be the associated extension group of π_{1}(Σ) by ⟨ψ⟩. Let O_{⟨ψ⟩} be the smallest subcover of Σ to which ⟨ψ⟩ descends, let C_{⟨ψ⟩} be the relative commensurator of ⟨ψ⟩ in M(O_{⟨ψ⟩}), and let Γ_{C_{⟨ψ⟩}} be the extension of π_{1}(O_{⟨ψ⟩}) by C_{⟨ψ⟩}. Note that C_{⟨ψ⟩} contains ⟨ψ⟩ with finite index by [BLM83], and so Γ_{C_{⟨ψ⟩}} contains Γ_{⟨ψ⟩} with finite index. The proofs of §7 now apply directly to obtain the following classification theorem, which parallels Theorem 1.1:

Theorem 8.2. Given closed surfaces Σ_{i}, i = 1, 2, and pseudo-Anosov mapping classes ψ_{i} of M(Σ_{i}), the following are equivalent, where O_{i} = O_{⟨ψ_{i}⟩}, etc.:

(1) There exists a horizontal respecting quasi-isometry Γ_{⟨ψ_{1}⟩} → Γ_{⟨ψ_{2}⟩}.

(2) The groups Γ_{⟨ψ_{1}⟩}, Γ_{⟨ψ_{2}⟩} are horizontal respecting commensurable, meaning that there is an isomorphism from a finite index subgroup H_{1} < Γ_{⟨ψ_{1}⟩} to a finite index subgroup H_{2} < Γ_{⟨ψ_{2}⟩}, taking H_{1} ∩ π_{1}(O_{1}) to H_{2} ∩ π_{1}(O_{2}).

(3) There is an isomorphism O_{1} ≈ O_{2} such that in the group M(O_{1}) = M(O_{2}) the mapping classes ψ_{1} and ψ_{2} have equal powers.

(4) There is an isomorphism O_{1} ≈ O_{2} such that in the group M(O_{1}) = M(O_{2}) the mapping classes ψ_{1} and ψ_{2} have the same limit set in the Thurston boundary of the Teichmüller space T (O_{1}) = T (O_{2}), i.e. they have the same stable/unstable measured foliation pairs.

(5) There is an isomorphism Γ_{C_{1}} ≈ Γ_{C_{2}} taking π_{1}(O_{1}) to π_{1}(O_{2}).

♢

It is easy to use this theorem to obtain infinitely many distinct horizontal respecting quasi-isometry classes of groups Γ_{H}, for cyclic pseudo-Anosov groups H. This should be contrasted with the fact that all of the groups Γ_{H} are quasiisometric to each other and to H^{3}, by Thurston’s hyperbolization theorem for fibered 3-manifolds [Ota96].

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Benson Farb: Department of Mathematics, Univeristy of Chicago 5734 University Ave. Chicago, Il 60637 E-mail: farb@math.uchicago.edu

Lee Mosher: Department of Mathematics, Rutgers University, Newark Newark, NJ 07102 E-mail: mosher@andromeda.rutgers.edu