Curvature and rank of Teichmüller space

Jeffrey Brock^{∗} and Benson Farb^{†}

September 6, 2001

Abstract

Let S be a surface with genus g and n boundary components and let d(S) = 3g − 3 + n denote the number of curves in any pants decomposition of S. We employ metric properties of the graph of pants decompositions C_{P}(S) prove that the Weil-Petersson metric on Teichmüller space Teich(S) is Gromov-hyperbolic if and only if d(S) ≤ 2. When d(S) ≥ 3 the Weil-Petersson metric has higher rank in the sense of Gromov (it admits a quasi-isometric embedding of ℝ^{k}, k ≥ 2); when d(S) ≤ 2 we combine the hyperbolicity of the complex of curves and the relative hyperbolicity of C_{P}(S) prove Gromov-hyperbolicity.

We prove moreover that Teich(S) admits no geodesically complete Gromov-hyperbolic metric of finite covolume when d(S) ≥ 3, and that no complete Riemannian metric of pinched negative curvature exists on Moduli space M(S) when d(S) ≥ 2.

1 Introduction

The Weil-Petersson metric on Teichmüller space Teich(S) has many curious properties. It is a Riemannian metric with negative sectional curvature, but its curvatures are not bounded away from zero or negative infinity. It is geodesically convex, but it is not complete. In this paper we show that in spite of exhibiting negative curvature behavior, the Weil-Petersson metric is not coarsely negatively curved except for topologically simple surfaces S. Our main theorem answers a question of Bowditch [Be, Question 11.4].

Theorem 1.1 Let S be a compact surface of genus g with n boundary components. Then the Weil-Petersson metric on Teich(S) is Gromov-hyperbolic if and only if 3g − 3 + n ≤ 2.

∗Research supported by the NSF.

†Research supported by the NSF and the Sloan Foundation.

The constant d(S) = 3g − 3 + n is fundamental in Teichmüller theory: it is the complex dimension of the Teichmüller space Teich(S), or more topologically, the number of curves in any pair-of-pants decomposition of S. An equivalent formulation of Theorem 1.1, then, is that the Weil-Petersson metric is Gromov-hyperbolic precisely when the interior int(S) is a torus with at most two punctures or a sphere with at most five punctures.

Theorem 1.1 exhibits the first example of a (Mod(S)-invariant) metric on a Teichmüller space of (real) dimension greater than 2 that is Gromovhyperbolic. In contrast, these Teichmüller spaces admit no complete Riemannian metric of pinched negative sectional curvature (Theorem 1.3 below). To summarize, the overlap of the positive and negative results in this paper give:

When d(S) = 2, the Weil-Petersson metic on Teich(S) is Gromov-hyperbolic, yet Teich(S) admits no (equivariant) complete, Riemannian metric with pinched negative curvature.

The fact that the Weil-Petersson metric is not Gromov-hyperbolic when d(S) ≥ 3 relies heavily on a geometric investigation of a combinatorial model for the Weil-Petersson metric constructed in [Br] (see below).

Constraints on metrics on M_{g,n}. In S. Kravetz’ 1959 paper [Kr], it was claimed that the Teichmüller metric on the moduli space M_{g,n} of Riemann surfaces of genus g ≥ 2 with n punctures is a complete metric with negative curvature in the sense of Busemann, i.e. that Teich(S_{g,n}) admits such an equivarient metric. The proof had an error, and the result was shown to be false in [Mas1]; in fact Masur-Wolf showed in [MW] that the Teichmüller metric is not even Gromov-hyperbolic (see [MP1] and [MP2] for two other proofs). The following theorem shows that this phenomenon has little to do with the Teichmüller metric; in fact it holds for a broad class of metrics.

We say that a geodesically complete metric space X has finite volume if for each ϵ > 0 there is no infinite collection of pairwise disjoint ϵ-balls embedded in X.

Theorem 1.2 Suppose d(S) ≥ 3. Then Teich(S) admits no proper, geodesically complete, Gromov-hyperbolic Mod(S)-equivariant path metric with finite covolume.

Theorem 1.2 applies in particular to the Teichmüller metric, which satisfies the finite volume condition, giving the main result of [MW] when d(S) ≥ 3. Theorem 1.2 also applies to McMullen’s Kähler hyperbolic metric h on Teich(S) constructed in [Mc] as it is complete and Riemannian, and has finite volume quotient on M_{g,n} (it is also quasi-isometric to the Teichmüller metric).

We note that our argument for Theorem 1.2 breaks down in the case of the Weil-Petersson metric, which is not complete as a metric space (and in particular not geodesically complete). Indeed, with the isometric action of Mod(S) on the Weil-Petersson metric, Dehn twists are infinite order elliptic elements, as they act with bounded orbits. The first part of our proof of Theorem 1.2 is essentially an argument of McCarthy-Papadopoulos [MP1], where the theorem is proven under the additional hypothesis that every pseudo-Anosov element acts as a hyperbolic isometry. We reproduce the argument here for completeness and since it is brief; we also extend the proof to the punctured case.

In the absence of the finite-volume hypothesis, it is possible to say something about complete Riemannian metrics of pinched negative curvature.

Theorem 1.3 If 3g −3+n ≥ 2 then M_{g,n} admits no complete Riemannian metric of pinched negative sectional curvature.

Note that Theorem 1.2 implies Theorem 1.3 in the finite volume case for d(S) ≥ 3 but does not cover the case when d(S) = 2, i.e. when int(S) is a torus with two punctures or a sphere with five punctures.

The rank of the Weil-Petersson metric. To prove that the Weil-Petersson metric is not Gromov-hyperbolic when d(S) ≥ 3 (the “only if” part of Theorem 1.1), we show it has higher rank in these cases.

The rank of a metric space X is the maximal dimension n of a quasi-flat in X, that is a quasi-isometric embedding ℝ^{n} → X. This notion of rank was introduced by Gromov ([Grom], 6.B_{2}), and agrees with the usual notion of rank of a nonpositively curved symmetric space (this follows easily from the quasi-flats theorem of [EF] and [KL]).

There has been a recurring comparison in the literature of Teichmüller space to symmetric spaces of noncompact type, particularly in terms of the rank one/higher rank dichotomy (see, e.g., [Iv1, Iv2, FLM]). The following theorem adds to the list of higher rank behavior of Teichmüller space.

Theorem 1.4 The rank of the Weil-Petersson metric on Teich(S) is at least d(S)/2.

As Gromov-hyperbolic metric spaces have rank one, Theorem 1.4 implies one direction of Theorem 1.1. We conjecture that the Weil-Petersson metric on Teich(S) has rank precisely the integer part of (d(S) + 1)/2; the conjecture is supported by the hierarchies machinery of [MM2], but bounding the rank from above appears delicate.

Remark. While Dehn twists about disjoint simple closed curves produce quasi-flats in the mapping class group (endowed with the word metric) - see [FLM], Dehn twist orbits in Teich(S) do not generate quasi-flats in either the Weil-Petersson or Teichmüller metrics: indeed, if τ ∈ Mod(S) is a Dehn twist, we have d_{WP}(X, τ^{n}X) = O(1) and d_{T}(X, τ^{n}X) = O(log(n)). Evidently, the appearance of an orbit of a subgroup generated by commuting Dehn twists in the Teichmüller metric is more akin to a horosphere in a rank-one symmetric space. (Note that in higher rank symmetric spaces horospheres are actually quasi-isometrically embedded.)

Combinatorics of curves on surfaces. The proof of Theorems 1.1 and 1.4 rely on important work of Masur and Minsky [MM1, MM2] on combinatorial complexes associated to curves on surfaces.

Let 𝒮 denote the set of all isotopy classes of essential, non-peripheral, simple closed curves on the surface S. The curve complex C(S) is the simplicial complex whose vertices are the elements of 𝒮 and whose k-simplices span collections of k + 1 curves in 𝒮 that can be realized pairwise disjointly on S. Metrizing each simplex to be the standard Euclidean simplex, one obtains a metric on C(S). The main theorem of [MM1] is that C(S) endowed with this metric is a Gromov-hyperbolic metric space.

Our proof of Theorem 1.4 uses a closely related combinatorial object. Consider the graph whose vertices are pair-of-pants decompositions of S, and whose edges join decompositions that differ by a single elementary move (see Figure 1). Assigning each edge length 1, we obtain a graph C_{P}(S) with a distance function d_{P}(., .) on pairs of vertices given by taking the minimal length path between two pants-decompositions. This graph is the 1-skeleton of a simplicial complex introduced by Hatcher-Thurston [HT]; in particular they proved that this graph is connected. The graph C_{P}(S) coarsely models Weil-Petersson geometry.

Theorem 1.5 ([Br]) The graph C_{P}(S) is quasi-isometric to Teich(S) endowed with the Weil-Petersson metric.

The proof of Theorem 1.4 can thus be reduced to finding quasi-isometrically embedded flats in the graph C_{P}(S). The hypothesis is explained by the need for at least two disjoint essential subsurfaces of S whose Teichmüller spaces are themselves non-trivial, in which case flats arise from pants decompositions related by elementary moves that occur in disjoint subsurfaces.

α α  α′ α′

Figure 1. Elementary moves on pants decompositions.

Conversely, when d(S) ≤ 2, one cannot perform independent elementary moves on pairs of pants because there are not enough disjoint subsurfaces on which to perform them. The lack of available subsurfaces leads one to consider the possibility that C_{P}(S) is Gromov-hyperbolicity in these cases. Gromov-hyperbolicity is known for d(S) = 1 by [MM1]; we establish Gromov-hyperbolicity of C_{P}(S) for for d(S) = 2 using three ingredients: hyperbolicity of the complex of curves C(S) (proved in [MM1]), the theory of relative hyperbolicity developed in [Fa], and the hierarchical structure of C(S) given in [MM2].

Plan of the paper. Section 2 gives preliminaries on Teichmüller theory and Weil-Petersson geometry, Gromov-hyperbolic metric spaces, and the combinatorics of curves on surfaces. Section 3 contains the proof of nonexistence of geodesically complete Gromov-hyperbolic metrics on Teich(S) with finite covolume when d(S) ≥ 3 (Theorem 1.2), and gives constraints on complete Riemannian metrics of pinched negative curvature on M_{g,n} (Theorem 1.3).

In section 4 we show that the Weil-Petersson metric has higher rank when d(S) ≥ 3 (Theorem 1.4), proving one direction of Theorem 1.1. Finally, in Section 5 we prove the other direction of Theorem 1.1 by showing the Weil-Petersson metric is Gromov-hyperbolic when d(S) ≤ 2.

We conclude the paper with a list of questions for further investigation.

Ackhowledgements. The authors would like to thank Howard Masur and Yair Minsky for many informative discussions on the topic of this paper.

2 Preliminaries

2.1 Teichmüller space

Let S be a topological surface, possibly with boundary. When S has boundary ∂S denote by int(S) its interior S −∂S. The Teichmüller space Teich(S) parameterizes finite area hyperbolic surfaces X equipped with markings, or homeomorphisms (f: int(S) → X) up to isometries that preserve the marking: i.e.

(f: int(S) → X) ∼ (g: int(S) → Y)

if there is an isometry ϕ: X → Y so that g ≃ ϕ ∘ f.

The space Teich(S) is topologized by the distance

d((f, X), (g, Y)) = inf_{φ} log L(φ)

where the infimum is taken over all bi-Lipschitz diffeomorphisms φ isotopic to g ∘ f^{−1} and L(φ) is the minimal bi-Lipschitz constant for φ. It is homeomorphic to a cell of dimension 6g−6+2n where n is the number of boundary components of S.

The Teichmüller space carries a natural complex structure; its complex cotangent space T_{X}^{∗}Teich(S) at X ∈ Teich(S) is identified with the space of holomorphic quadratic differentials Q(X) on X. The Weil-Petersson metric on Teich(S) is obtained by duality from the L^{2}-inner product on Q(X)

⟨φ, ψ⟩_{WP} = (∫_{X} \frac{φ\overline{ψ}}{ρ^{2}} |dz|^{2})^{1/2}

where ρ(z)|dz| is the hyperbolic line element on X.

2.2 The pants complex

Let 𝒮 denote the isotopy classes of essential simple closed curves on S. A pants decomposition P of S is a maximal collection of distinct elements of 𝒮 so that no two isotopy classes in P have representatives that intersect. The pants complex C_{P}(S) is the graph with one vertex for each pants decomposition and an edge joining each pair of vertices whose pants decompositions differ by an elementary move (see figure 1). The distance function

d_{P} : C_{P}^{0}(S) × C_{P}^{0}(S) → ℤ_{≥0}

on the vertex set C_{P}^{0}(S) of C_{P}(S) counts the minimal number of elementary moves between maximal partitions.

Geodesic representatives. We briefly describe the quasi-isometry of Theorem 1.5.

Given X ∈ Teich(S), the elements of P can be represented as pairwise disjoint closed geodesics on X. The sub-level sets for the lengths of the elements of P play a special role in Weil-Petersson geometry. Let

V_{L}(P) = {X ∈ Teich(S) | ℓ_{X}(α) < L for each α ∈ P } .

Applying theorems of Wolpert [Wol] and Masur [Mas2], we have the following (see [Br]):

Proposition 2.1 Given L there is a D_{L} so that for each P ∈ C_{P}^{0}(S), the sub-level set has Weil-Petersson diameter

diam_{WP}(V_{L}(P)) < D_{L}.

By a theorem of Bers, there is an L > 0 so that the union

⋃_{P ∈C_{P}^{0}(S)} V_{L}(P)

covers Teich(S). Let V (P) = V_{2L}(P). Let

Q: C_{P}^{0}(S) → Teich(S)

be any map for which Q(P) ∈ V (P). Then Theorem 1.1 of [Br] shows Q is a quasi-isometry. In other words there are constants K_{1} > 1 and K_{2} > 0 depending only on S so that given X ∈ V (P_{X}) and Y ∈ V (P_{Y}), we have

\frac{1}{K_{1}} d_{P}(P_{X}, P_{Y}) − K_{2} ≤ d_{WP}(X, Y) ≤ d_{P}(P_{X}, P_{Y}) + K_{2}.

2.3 The curve complex

Let C(S) be the complex associated to the simple closed curves 𝒮 on S as follows:

• The zero-skeleton C^{0}(S) is identified with the elements of 𝒮.

• Any k + 1 curves (α_{1}, …, α_{k+1}) in 𝒮^{k+1} with the property α_{i} ≠ α_{j} and i(α_{i}, α_{j}) = 0, for i ≠ j determine a k simplex in C(S).

For an essential, non-annular subsurface Y ⊂ S, the curve complex C(Y) is a subcomplex of C(S). The subsurface projection

π_{Y} : C(S) → P(C(Y))

from the curve complex C(S) to the set P(C(Y)) of all subsets of C(Y) is defined by setting π_{Y} (α) = α if α ∈ C(Y) and letting

π_{Y} (α) = ∪_{α′}∂N (α′ ∪ ∂_{α′}Y)

be the union over all arcs α′ of essential intersection of α with Y of a regular neighborhood ∂N (α′ ∪ ∂_{α′}Y) of α′ ∪ ∂_{α′}Y, where ∂_{α′}Y represents the components of the boundary of Y that α′ intersects (see [MM2, Sec. 2]).

For two subsets A and B of C(Y), the (semi)-distance d_{Y} (A, B) is defined by

d_{Y} (A, B) = diam_{C(Y)}(A ∪ B).

This notion of distance allows us to measure the distance between pants decompositions P and P′ relative to Y by letting π_{Y} (P) = ∪_{α∈P} π_{Y} (α) and letting the projection distance d_{Y} (P, P′) between P and P′ relative to Y be

d_{Y} (P, P′) = d_{Y} (π_{Y} (P), π_{Y} (P′)).

We also record for reference the following Lipschitz property for the projection π_{Y}:

Lemma 2.2 (Lem. 2.3 of [MM2]) Let Y be an essential subsurface of S. Then if α and β simple closed curves whose vertices have distance one in C(S) and π_{Y} (α) ≠ ∅ ≠ π_{Y} (β), then we have d_{Y} (α, β) ≤ 2.

For a detailed discussion of the curve complex and related projection mappings, see [MM1] and [MM2]

2.4 Hyperbolic metric spaces

In this subsection we briefly recall some material on hyperbolic metric spaces. The standard reference for this material is [GH]. All statements about hyperbolic metric spaces which we use can be found in [GH].

A metric space X is proper if every closed ball in X is compact. If for any x, y ∈ X, there exists a parametrized path γ : [0, d(x, y)] → X from x to y with d(γ(s), γ(t)) = |s − t| for all s, t ∈ [0, d(x, y)] then X is called a geodesic metric space. Here we are using the metric space notion of path length.

X is a Gromov-hyperbolic, or δ-hyperbolic metric space if there exists δ > 0 so that every geodesic triangle T in X is δ-thin: the δ-neighborhood of any two sides of T contains the third side.

A δ-hyperbolic metric space has a natural compactification X∪∂X where ∂X is the collection of Hausdorff equivalence classes of geodesic rays in X. Every isometry of X acts by homeomorphisms on ∂X. We denote the fixed set of the action of an isometry g ∈ Isom(X) in ∂X by Fix(g).

Combining Theorems 16 and 17 of Section 8 of [GH] gives the following classification theorem.

Theorem 2.3 (Classification of isometries) Every isometry g of a δhyperbolic metric space is precisely one of the following types:

1. elliptic: every g-orbit is bounded.

2. hyperbolic: Fix(g) = {x, y} for some x ≠ y in ∂X, in which case any g-orbit in X is a quasi-geodesic with limit set {x, y}.

3. parabolic: Fix(g) = {x} for some x ∈ ∂X.

2.5 The rank of a metric space

A metric space has a quasi-flat of dimension n if there is a quasi-isometric embedding F: ℝ^{n} → X. We say a metric space has higher rank if it admits a quasi-flat of dimension at least 2. Since ℝ^{n} is not Gromov-hyperbolic, the quasi-isometric embedding F provides a family of triangles in X that violates the δ-thin condition for all δ > 0. Thus, a higher rank metric space is not Gromov-hyperbolic.

3 Constraints on metrics on M_{g,n}

In this section we prove Theorems 1.2 and 1.3.

The idea of Theorem 1.2 is that a properly discontinuous action of Mod(S) on a geodesically complete, δ-hyperbolic metric space has certain special properties when S is sufficiently complicated. Indeed, when d(S) ≥ 3, then entire group Mod(S) must act parabolically with a single parabolic fixed point at infinity. This first part is essentially an argument of McCarthy-Papadopoulos [MP2]. As with isometric actions on hyperbolic space, we show that such an action cannot have finite volume quotient.

Notation: For notational purposes in the following arguments, we will use S_{g,n} to refer to a surface with genus g and n boundary components.

Proof: (of Theorem 1.2). Suppose to the contrary that X = Teich(S_{g,n}) admits a Mod(S_{g,n})-equivariant, geodesically complete path metric which is δ-hyperbolic. We have that Mod(S_{g,n}) acts properly discontinuously on X by isometries, and (see §2.4) that Mod(S_{g,n}) thus acts by homeomorphisms on the Gromov boundary ∂X.

We claim that since 3g − 3 + n ≥ 3, we may pick a generating set {g_{i}} for Mod(S_{g,n}) consisting of Dehn twists about non-separating curves, with the following properties:

1. Each g_{i} is conjugate in Mod(S_{g,n}) to each g_{j}.

2. The group generated by elements commuting with g_{1} is not virtually cyclic, i.e. it does not contain a cyclic subgroup of finite index; similary for g_{2}.

3. g_{1} and g_{2} do not commute; in fact sufficiently high powers of g_{1} and g_{2} generate a free group.

4. The commuting graph for {g_{i}}, consisting of a vertex for each g_{i} and an edge connecting commuting elements, is connected.

When n = 0 one may take the “standard” Dehn-Lickorish-Humphries generators (see [FF]). For n > 0 one proceeds inductively by using the exact sequence (see, e.g. [Iv2]):

1 → π_{1}(S_{g,n}) → Mod(S_{g,n+1}) → Mod(S_{g,n}) → 1

where the kernel is generated by “pushing the puncture” around a given loop, one for each loop in a standard generating set for π_{1}(S_{g,n}). Such elements of Mod(S_{g,n+1}) are generated by elements α_{1}α_{2}^{−1} where each α_{i} is a Dehn twist about a non-separating curve; one then adds these Dehn twists to the previous list of generators, easily checking the required properties, and continues inductively.

The elements g_{1} and g_{2} may then be taken to be an intersecting pair of loops in the Dehn-Humphries-Lickorish generating set. In particular, the group generated by elements commuting with g_{i} (for i = 1 or i = 2), contains the mapping class group Mod(S_{g,n−1}), which is not virtually cyclic for 3g − 3 + n ≥ 3.

Now apply the classification of isometries of δ-hyperbolic metric spaces (see Theorem 2.3 above) to g_{1}. Note that this classification uses the geodesic assumption on the metric space X. As g_{1} has infinite order and the action of Mod(S_{g,n}) is properly discontinuous, it follows that g_{1} is not of elliptic type. Suppose g_{1} is of hyperbolic type. Then Fix(g_{1}) = {x, y} for some x ≠ y in ∂X.

The subgroup H of Mod(S_{g,n}) commuting with g_{1} clearly leaves {x, y} invariant, and is not virtually cyclic. But Theorem 30 in Section 8 of [GH] states that, for a group Γ acting properly discontinuously on a proper, geodesic, δ-hyperbolic metric space X, the stabilizer of a pair of distinct points {x, y} has a cyclic subgroup of finite index, a contradiction. Thus it must be that g_{1} is of parabolic type, and so Fix(g_{1}) = x for some x ∈ ∂X.

As each g_{i} is conjugate to g_{1}, each g_{i} is also of parabolic type, say fixing the unique point x_{i} ∈ ∂X. As Dehn twists about disjoint curves commute, since [g, h] = 1 implies g(Fix(h)) = Fix(h), and since the commuting graph of {g_{i}} is connected, it follows that Fix(g) = x for each g ∈ {g_{i}}.

We claim that every element of Mod(S_{g,n}), not just the generating set, is of parabolic type with unique fixed point x ∈ ∂X. As Mod(S_{g,n}) has a torsion-free subgroup of finite index, if this is not true then there exists g ∈ Mod(S_{g,n}) acting on X as an isometry of hyperbolic type, with x as an attracting point. Pick any z ∈ X. By Theorem 8.21 of [GH], the orbit {g^{n}z : n ∈ ℤ} is a K-quasigeodesic in x for some K ≥ 1 and has limit point x as n → ∞. Pick any element h ∈ Mod(S_{g,n}) acting as a parabolic fixing x (such h exist by the previous paragraph). Since {g^{n}z : n ∈ ℤ} and {hg^{n}z : n ∈ ℤ} are both K-quasigeodesics which limit to x as n → ∞, it follows that there exists N > 0, C > 0 so that d(g^{n}(z), hg^{n}(z)) < C for all n > N. But then d(g^{−n}hg^{n}(z), z) = d(hg^{n}(z), g^{n}(z)) < C for all n > N. Since h is of parabolic type and g is of hyperbolic type, the set {g^{−n}hg^{n} : n ∈ ℤ} is infinite, so that the hypothesis of proper discontinuity is violated. Hence the claim is proved ^{1}.

It follows that Mod(S_{g,n}) permutes the set of equivalence classes of geodesic rays with x as their common endpoint at infinity x ∈ ∂X. We now show that one of the hypotheses of the theorem must be violated.

Recall from §8.1 of [GH] that any choice of a point x ∈ ∂X and basepoint y ∈ X determines a Busemann function β : X → ℝ on X defined by

β(z) = sup_{γ}{lim sup_{t→∞}(d_{X}(z, γ(t)) − t))}

where the sup is taken over all geodesic rays γ based at y with γ(∞) = x. In the proof of Proposition 8.18 of [GH] (see also Remark 8.13.ii) it is shown that any parabolic isometry g fixing x ∈ ∂X must almost preserve level sets of the Busemann function, that is there exists a constant C so that |β(w) − β(g^{n}(w))| ≤ C for all n ∈ ℤ and any w lying on the ray γ.

1While this claim is obviously true in the negatively curved Riemannian context without the proper discontinuity hypothesis, it may not be true without it for arbitrary Gromov-hyperbolic spaces; see [GH], 8.13.

As this holds true for all g ∈ Mod(S_{g,n}), it follows that any Mod(S_{g,n})- orbit in X lies within a bounded distance of some level set of β. But β is clearly proper on γ; in particular there exists a constant ϵ and points z_{i}, i = 1, 2, … on γ with the property that for any i, j with i ≠ j, the ϵ-ball centered at z_{i} is disjoint from the Mod(S_{g,n})-orbit of the ϵ-ball centered at z_{j}. In particular the quotient Teich(S_{g,n})/Mod(S_{g,n}) contains an infinite, disjoint collection of ϵ-balls. This contradicts the finite volume hypothesis.

□

For Theorem 1.3, we illustrate that the existence of a Mod(S)-equivariant complete Riemannian metric on Teich(S) of pinched negative curvature puts even stronger restrictions on an isometric action of Mod(S).

Proof: (of Theorem 1.3). If M_{g,n} did admit such a metric, then lifting this metric to the universal cover Teich(S_{g,n}) gives a properly discontinuous, isometric action of Mod(S_{g,n}) on a complete, 1-connected, pinched negatively curved manifold X = Teich(S_{g,n}).

Since 3g − 3 + n ≥ 2, there exists a subgroup N of Mod(S_{g,n}) generated by Dehn twists as above, with elements conjugate in Mod(S_{g,n}), and with the property that N is not virtually nilpotent (since it contains, for example, noncyclic free subgroup (see, e.g. [FLM])). Note that in case g = 0, the elements g_{i} can be taken to be Dehn twists about curves which surround two punctures (hence are separating), in particular they are conjugate in Mod(S_{g,n}).

As X is δ-hyperbolic, the exact same argument as in the proof of Theorem 1.2 gives some x ∈ X which is fixed by each generator g_{i}.

Fix any horosphere H based at x, and fix a basepoint s ∈ H. Then d(s, g_{i}s) ≤ C_{H} for some constant C_{H} for each generator g_{i}. As the sectional curvature of X is pinched between two negative constants, we may find a horosphere H based at x ∈ ∂X so that C_{H} is as small as we want; the key point is that the pinching of the curvatures gives a definite (exponential) rate at which geodesic rays asymptotic to x converge. Choosing H so that C_{H} is smaller than the Margulis constant for X (which depends only on dim(X) and on the pinching constants of the sectional curvatures) and applying the Margulis Lemma (see, e.g. [BGS]), it follows that N contains a nilpotent subgroup of finite index, a contradiction. □

Remark. Theorem 1.3 may also be deduced (though not in some of the low genus cases) from the topological structure of the end of M_{g,n}. On the one hand, every end of a finite volume manifold of pinched negative curvature is homeomorphic to the product of a compact nilmanifold and [0, ∞); this is essentially the Margulis Lemma (see [BGS]). On the other hand, it seems to be well-known (see, e.g. [Fa]) that the entire orbifold fundamental group of M_{g,n} is carried by its end; in particular the fundamental group of the end is not virtually nilpotent. This argument actually shows that no finite cover of M_{g,n} admits a complete, finite volume Riemannian metric of pinched negative curvature.

4 Quasi-flats in the pants complex

Let S be a surface of genus g with n boundary components. A subsurface R ⊂ S is a compact connected embedded surface lying in S. The subsurface R is essential if its boundary components are homotopically essential in S.

We say S decomposes into essential subsurfaces R_{1}, …, R_{k} if each R_{j} may be modified by an isotopy so that they are pairwise disjoint and S − R_{1} ⊔ … ⊔ R_{k} is a collection of open annular neighborhoods of simple closed curves on S, each isotopic to a boundary component of R_{j}.

Let r(S) denote the maximum number k in any decomposition of S into essential subsurfaces R_{1}, …, R_{k} such that each R_{j}, j = 1, …, k has genus at least one, or at least four boundary components. Then r(S) is greatest integer less which is at most (d(S) + 1)/2.

Theorem 4.1 The complex C_{P}(S) contains a quasi-flat of dimension r(S).

Proof: The surface S decomposes into subsurfaces

R_{1}, …, R_{r(S)}, T

so that d(R_{j}) = 1 for each j and either T is empty or d(T) = 0. Let n = r(S). We describe a quasi-isometric embedding of the Cayley graph for ℤ^{n} with the standard generators into the complex C_{P}(S).

Let c_{j} be a vertex in the curve complex C(R_{j}). Then together with the core curves of the open annuli in S − R_{1} ⊔ … ⊔ R_{r(S)} ⊔ T, the curves c_{j} form a pants decomposition P = P (c_{1}, …, c_{n}) of S.

We let g_{j}: ℤ → C(R_{j}) be a geodesic so that g_{j}(0) = c_{j}. Note that C(R_{j}) is the Farey graph, so we may take any bi-infinite geodesic g_{j} in C(R_{j}).

We claim that the embedding

Q: ℤ^{n} → C_{P}(S)

defined by

Q(k_{1}, …, k_{n}) = P (g_{1}(k_{1}), …, g_{n}(k_{n}))

is a quasi-isometry, whose constants depend only on S.

Let k⃗ = (k_{1}, …, k_{n}) and l⃗ = (l_{1}, …, l_{n}). Since elementary moves along g_{j} can be made independently in each R_{j}, we have

d_{P}(Q(k⃗), Q(l⃗)) ≤ ∑_{j=1}^{n} |l_{j} − k_{j}| = d_{ℤ^{n}}(k⃗, l⃗)

which shows that Q is 1-Lipschitz.

Given R_{j}, the projection π_{R_{j}}(Q(k⃗)) to R_{j} described in §2.3 simply picks out the curve g_{j}(k_{j}) so we have

π_{R_{j}}(Q(k⃗)) = g_{j}(k_{j}).

Thus, the projection distance

d_{R_{j}}(Q(k⃗), Q(l⃗)) = d_{R_{j}}(g_{j}(k_{j}), g_{j}(l_{j}))

which is simply |k_{j} − l_{j}| since g_{j} is a geodesic in C(R_{j}).

By Theorem 6.12 of [MM2], there exists M_{0} = M_{0}(S) so that for all M ≥ M_{0} there exist constants K_{0} and K_{1} so that if we let Pk⃗ = Q(k⃗) and Pl⃗ = Q(l⃗) then we have the inequality

∑_{d_{Y} (π_{Y} (Pk⃗),π_{Y} (Pl⃗))>M^{Y ⊆S}} d_{Y} (Pk⃗, Pl⃗) ≤ K_{0}d_{P}(Pk⃗, Pl⃗) + K_{1}.

But the left-hand-side of the inequality is bounded below by

max_{j} |k_{j} − l_{j}| ≥ \frac{∑_{j} |k_{j} − l_{j}|}{n}.

Thus, Q is a quasi-isometric embedding. □

5 The case of low genus

In the case where d(S) = 1, where int(S) is homeomorphic to a punctured torus or four-times-punctured sphere, the pants complex C_{P}(S) is identified with the curve complex C(S), which is Gromov-hyperbolic (see [MM1]).

Together with the previous section, this observation leaves one case unattended, namely that when d(S) = 2. In this case, int(S) is homeomorphic either to a doubly-punctured torus, or a five-times-punctured sphere. In this section we prove the following.

Theorem 5.1 Let S be such that d(S) equals 1 or 2. Then the Weil-Petersson metric on Teich(S) is Gromov-hyperbolic.

The case d(S) = 1 is proven in [MM1], since C(S) = C_{P}(S) in this case. Thus we are left to treat the case when d(S) = 2. In this case, we apply results of the second author which, although initially phrased in the context of groups with their word metrics, apply to general metric spaces.

Our argument will employ the notion of relative hyperbolicity developed in [Fa]. In essence, a metric space is relatively hyperbolic relative to a collection of subsets if the result of crushing those subsets to have diameter 1 is a hyperbolic metric space. In the case d(S) = 2, we will show that C_{P}(S) is relatively hyperbolic relative to regions consisting of pants decompositions containing a single curve. Since, in this case, such regions are themselves hyperbolic, it is possible to establish Gromov-hyperbolicity C_{P}(S) by showing that paths in C_{P}(S) that determine quasi-geodesics in the relative space satisfy certain boundedness properties with respect to their trajectories through these regions (this is the bounded region penetration property, below).

Remark: One can prove theorem 5.1 by directly demonstrating a thintriangles condition after replacing geodesics in C_{P}(S) by the hierarchies of [MM2] (this was our original approach to the argument). We have chosen instead to employ the theory relative hyperbolicity as it is more familiar, and unifies these cases with the higher genus cases. Indeed, when d(S) > 2 the natural regions with respect to which C_{P}(S) is relatively hyperbolic (sub-graphs of pairs of pants containing a given curve α) are not themselves hyperbolic; they are the quasi-flats of the previous section.

Relative hyperbolicity. Let (X, d) be a geodesic metric space, and let H_{α} be a collection of connected subsets of X, with index α in an index set A. Then the electric space X̂ relative to the regions {H_{α}} is obtained by collapsing each region to have diameter one, as follows (see [Fa, Sec. 3]). Adjoin to the space X a single point c_{α} for each α ∈ A by connecting c_{α} to each point of H_{α} by a segment of length 1/2. Let X̂ denote the resulting path-metric space and let d_{e}(., .) denote path distance in X̂. Given a path w in X we obtain a path in X̂ by replacing segments where w travels in H_{α} with a path joining the endpoints of the segment to c_{α}. As in [Fa], we denote this path-replacement procedure by X → X̂ or w ↦ ŵ. We denote by I(w) the initial point of w and by T (w) the terminal point of w. The points I(w) and T (w) depend on the choice of parameterization.

If ŵ is a (k-quasi) geodesic in X̂ we say w is a relative (k-quasi) geodesic in X. If a path w in X (or ŵ in X̂) passes through some region H_{α} we say it penetrates H_{α}. A path w ∈ X (or ŵ in X̂) has no backtracking if for every region H_{α} that ŵ penetrates, once it leaves H_{α} it never returns. The space X is hyperbolic relative to {H_{α}}_{α∈A} if the electric space X̂ is Gromov hyperbolic.

In the pants complex, consider following collection of regions: for each α ∈ C(S) let

H_{α} = {P ∈ C_{P}(S) | α ∈ P }.

Then we have the following theorem (cf. [MM1, Thms. 1.2, 1.3]).

Lemma 5.2 The graph C_{P}(S) is hyperbolic relative to the regions {H_{α}}.

Proof: It suffices to show that the electric space { \atop C_{P}(S)} with respect to the regions {H_{α}} is quasi-isometric to the curve complex C(S), which is Gromov-hyperbolic by [MM1, Thm. 1.1].

To see this, let Γ = C_{P}(S), let Γ̂ be the electric space associated to the regions {H_{α}}, and let c_{α} be the point added to Γ at distance 1/2 from each point of H_{α} to form Γ̂. Consider the mapping

q: C^{0}(S) → Γ̂

from the zero-skeleton of C(S) to Γ obtained by setting q(α) = c_{α}. Note that given a pants decomposition P, if β is an element of P then P lies a distance 1/2 from c_{β}, so the image q(C^{0}(S)) is 1/2-dense.

Moreover, we have d_{C(S)}(α, β) = 1 if and only if there is a P for which α ∈ P and β ∈ P. But α ∪ β ⊂ P holds if and only if the regions H_{α} and H_{β} intersect, which holds if and only if

dΓ̂(c_{α}, c_{β}) = 1.

Thus, the map q is 1-bi-Lipschitz, and since the image is 1/2-dense we may construct a 2-Lipschitz inverse to q. Thus, q a quasi-isometry. □

Bounded region penetration. We now recall results of [Fa] detailing a criterion on relative quasi-geodesics that will serve to ensure hyperbolicity of C_{P}(S) when d(S) = 2. The following definition is analogous to the “bounded coset penetration property” in §3.3 of [Fa].

Definition 5.3 (bounded region penetration) The pair (X, {H_{α}}) satisfies the bounded region penetration property if, for every P ≥ 1 there is a constant c = c(P) > 0 so that if u and w are relative P -quasi-geodesics without backtracking so that the initial and terminal points of u and w satisfy d_{X}(I(u), I(w)) ≤ 1 and d_{X}(T (u), T (w)) ≤ 1 then the following holds:

1. If u penetrates a region H_{α} but w does not penetrate H_{α}, then u travels an X-distance at most c in H_{α}.

2. If both u and w penetrate a region H_{α} then the points at which u and w first enter H_{α} lie an X-distance at most c from one another and likewise for the exit points.

An important application is the following theorem in which bounded region penetration is used to bootstrap from hyperbolicity relative to hyperbolic regions to hyperbolicity of the original metric space.

Theorem 5.4 Suppose X is hyperbolic relative to the regions {H_{α}} and that the pair (X, {H_{α}}) has the bounded region penetration property. Then if the regions H_{α} are themselves δ-hyperbolic metric spaces for some δ > 0, then X is a Gromov-hyperbolic metric space.

Proof: (of Theorem 5.4). Theorem 5.4 is simply a recasting of the remark following [Fa, Thm. 3.8] from groups to general metric spaces. The proof works verbatim in this case, with the following addition: one replaces the use of the theorem that linear isoperimetric inequality for a group implies that the group is Gromov-hyperbolic by the corresponding theorem for metric spaces with a well-defined notion of area. Such a theorem is proven by Bowditch in [Bowd]; in this case one can use for area the combinatorial area of the simplicial complex whose 1-skeleton is the pants graph C_{P}(S) and whose 2-cells consist of five types of loops with 3, 4, 5, and 6 edges, and no other edges between vertices (this is a variant of the 2-complex studied by Hatcher-Thurston in [HT]. It is shown to be simply connected in [HLS, Thm. D]). One may verify that Bowditch’s proof extends to the locally infinite case. □

Proof: (of Theorem 5.1). The condition d(S) = 2 implies that each pants decomposition of S is built from exactly two disjoint simple closed curves on S.

Lemma 5.5 Suppose d(S) = 2. Then there is a δ so that for each α ∈ C(S), the region H_{α} is δ-hyperbolic.

Proof: Given α ∈ C(S), let Y_{α} denote the connected component of the complement of an embedded open annular neighborhood of α for which d(Y_{α}) = 1. Then the region H_{α} is isometric to the curve complex C(Y_{α}). Again, [MM1, Thm. 1.1] implies that H_{α} is hyperbolic. □

To prove Theorem 5.1, then, it suffices to prove that when d(S) = 2, the pair (C_{P}(S), {H_{α}}) has the bounded region penetration property. To this end, let u and w be two relative P -quasi-geodesics in C_{P}(S) without backtracking, so that d_{P}(I(u), I(w)) ≤ 1 and d_{P}(T (u), T (w)) ≤ 1. Let H_{α} be a region which u penetrates but w does not. Being relative P -quasi-geodesics in the relatively hyperbolic space (C_{P}(S), {H_{α}}), it follows form the definition that the projections û and ŵ D-fellow-travel in the electric space { \atop C_{P}(S)} for some D > 0 depending only on P. For simplicity of notation let Γ = C_{P}(S) and let Γ̂ be the associated electric space relative to the regions {H_{α}}.

We observe that in our circumstances, the path replacement u ↦ û can be viewed as producing an explicit path in C(S) from a path u in Γ. Since d(S) = 2, each pants decomposition in of S has two elements, so a path u in Γ is a sequence of edges in C(S) each of which is joined to the previous one at one of its two endpoints. Let ũ denote the path in C(S) obtained by removing all but the first and last edges in any sequence of consecutive edges in u that all contain a single vertex. The path ũ in C(S) has image û under the quasi-isometry q (up to segments of length 1/2 at the endpoints).

To employ the extra information the curve complex provides, we work with ũ and w̃ rather than û and ŵ. The condition that u is a relative P -quasi-geodesic without backtracking simply means that the path ũ is a P -quasi-geodesic in C(S) that never repeats a vertex. Proving that (Γ, {H_{α}}) satisfies bounded region penetration property, then, reduces to verifying that for paths u and w in Γ whose for which d_{Γ}(I(u), I(w)) ≤ 1 and d_{Γ}(T (u), T (w)) ≤ 1, and whose corresponding paths ũ and w̃ are P -quasi-geodesics without backtracking we have:

1′ If ũ encounters a vertex v_{α} that w̃ avoids, the vertices v_{β} and v_{γ} adjacent to v_{α} on ũ have distance

d_{Y_{α}}(v_{β}, v_{γ}) < c

in the subsurface Y_{α}.

2′ If ũ and w̃ each encounter a vertex v_{α}, then the vertices v_{β} ∈ ũ and v_{β′} ∈ w̃ just prior to the encounter with v_{α} satisfy

d_{Y_{α}}(v_{β}, v_{β′}) < c

and likewise for the points v_{γ} and v_{γ′} on ũ and w̃ just following the encounter with v_{α}.

(For the remainder of this section we will denote by v_{α} the vertex in C(S) corresponding to the simple closed curve α on S to avoid notational confusion). To see property (1′) implies property (1) above, note that the condition that {v_{β}, v_{α}, v_{γ}} is a sub-segment of ũ implies that P = {v_{α}, v_{β}} and P′ = {v_{α}, v_{γ}} are the pants decompositions along the path u where u enters and exits the region H_{α}. If d_{Y_{α}}(v_{β}, v_{γ}) < c then there is a sequence of pants decompositions joining P to P′ of length at most c given by taking a geodesic

{v_{β} = v_{0}, …, v_{N} = v_{γ}}

joining v_{β} to v_{γ} in C(Y_{α}) and taking the sequence of pants decompositions to be {P_{j} = {v_{α}, v_{j}}}_{j}. One argues similarly that property (2′) implies property (2).

We now verify that properties (1′) and (2′) hold. Since ũ does not backtrack, we may choose points x and y on ũ on either side of the vertex v_{α} as follows. Either x is an endpoint of ũ or x lies at distance 2D in C(S) from the vertex v_{β} adjacent on ũ to v_{α} whichever is closer along ũ. The point y is either an endpoint of ũ or y lies at distance 2D along ũ from the vertex v_{γ} adjacent to v_{α} whichever is closer along ũ. There is then a path p on w̃ (which does not encounter v_{α}) from the nearest point to x on w̃ to the nearest point to y. Letting p_{x} be the shortest path joining x to w̃ and letting p_{y} be the shortest path joining y to w̃ in C(S), the concatenation

q = p_{x} ∘ p ∘ p_{y}

is a path in C(S) that avoids the vertex v_{α}. Moreover, the path q has length at most 8D.

Letting q_{x} be the path along ũ joining x to v_{β}, and letting q_{y} be the path along ũ joining y to v_{γ}, we have a path

r = q_{x} ∘ p ∘ q_{y}

of length at most 12D that avoids the vertex v_{α} entirely.

The path r describes a path in the curve complex C(S) so that each vertex along the interior of r corresponds to a curve that either lies in C(Y_{α}) or intersects ∂Y_{α}. It follows from lemma 2.2 that any two consecutive vertices z and z′ on r satisfy

d_{Y_{α}}(z, z′) ≤ 2.

Thus we have the bound

d_{Y_{α}}(v_{α}, v_{β}) ≤ 24D.

A similar argument proves property (2) of the bounded region penetration property holds. Choose a point x on ũ so that either x is the first vertex of ũ or x is at distance 2D along ũ from v_{β}, whichever is closer along ũ to v_{β}. By fellow traveling, there is a point y on w̃ and a path p of length at most D joining x to y in C(S). The path along w̃ joining y to the last vertex v_{η} prior to v_{α} along w̃ has length at most 2D(1 + P) in C(S), so there is a path r in C(S) of total length bounded by 3D + 2D(1 + P) joining v_{β} to v_{η} that does not hit v_{α} in its interior. Again, by lemma 2.2 any two consecutive vertices z and z′ on r have the property that

d_{Y_{α}}(z, z′) ≤ 2,

so we have

d_{Y_{α}}(v_{α}, v_{η}) ≤ 2(3D + 2D(1 + P)).

The same argument proves that points on v′_{α} and v′_{η} adjacent to v_{α} where ũ and w̃ depart from v_{α} also have bounded distance d_{Y_{α}}(v′_{α}, v′_{η}).

Having verified that properties (1′) and (2′) hold, we conclude that the pair (Γ, {H_{α}}) has the bounded region penetration property. The theorem follows from theorem 5.4. □

6 Questions

We close the paper with some natural questions.

Question 6.1 (McMullen) Does M_{g,n} admit a complete, nonpositively curved Riemannian metric?

McMullen’s Kähler metric on M_{g,n} is complete but not nonpositively curved, while the Weil-Petersson metric is nonpositively curved but is not complete. Is there a possible compromise?

Question 6.2 What is the geometric rank of

• the Weil-Petersson metric?

• the Teichmüller metric?

• the mapping class group?

Theorem 1.4 gives the lower bound d(S)/2 to the rank of the Weil-Petersson metric, while [Min] and [FLM] establish the lower bound d(S) for the rank of the Teichmüller metric and the mapping class group respectively.

The answers to these rank questions seem to be essential to understanding quasi-isometric rigidity questions in Teichmüller space and the mapping class group.

Question 6.3 If int(S) is homeomorphic to a doubly-punctured torus or 5times-punctured sphere, are the sectional curvatures of the Weil-Petersson metric bounded away from zero?

Were the (geodesically convex) Weil-Petersson metric to have curvature pinched from above by a negative constant, its Gromov-hyperbolicity in this case would be an immediate consequence.

References

[BGS] W. Ballman, M. Gromov, and V. Schroeder. Manifolds of Nonpositive Curvature. Birkhauser, 1985.

[Be] M. Bestvina. Questions in Geometric Group Theory. Available

        online at http://www.math.utah.edu/~bestvina.

[Bowd] B. H. Bowditch. Notes on Gromov’s hyperbolicity criterion. In E. Ghys, A Haefliger, and A. Verjovsky, editors, Group Theory from a Geometrical Viewpoint, pages 64–167. World Scientific, 1991.

[Br] J. Brock. The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores. Preprint, 2001.

[EF] A. Eskin and B. Farb. Quasi-flats and rigidity in higher rank symmetric spaces. Journal Amer. Math. Soc. 10(1997), 653–692.

[Fa] B. Farb. Relatively hyperbolic groups. Geom. & Funct. Anal. 8(1998), 1–31.

[Fa] B. Farb. A primer on mapping class groups. In preparation.

[FF] B. Farb and J. Franks. Groups of homeomorphisms of onemanifolds, I: actions of nonlinear gropus. Preprint (2001).

[FLM] B. Farb, A. Lubotzky, and Y. Minsky. Rank one phenomena for mapping class groups. Duke Math. J. 106(2001), 581–597.

[GH] E. Ghys and P. de la Harpe, editors. Sur les groupes hyperboliques d’ápres Mikhael Gromov, volume 83 of Progress in Mathematics. Birkhauser, 1990.

[Grom] M. Gromov. Asymptotic invariants of infinite groups. In G. Niblo and M. Roller, editors, Geometric Group Theory, volume 182 of LMS Lecture Notes. Cambridge Univ. Press, 1993.

[HLS] A. Hatcher, P. Lochak, and L. Schneps. On the Teichmüller tower of mapping class groups. J. reine angew. Math. 521(2000), 1–24.

[HT] A. Hatcher and W. Thurston. A presentation for the mapping class group. Topology 19(1980), 221–237.

[Iv1] N. V. Ivanov. Subgroups of Teichmüller Modular Groups, volume 115 of AMS Translations of Math. Monographs. American Mathematical Society, 1992.

[Iv2] N. V. Ivanov. Mapping class groups. Preprint.

[KL] B. Kleiner and B. Leeb. Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings. Publ. IHES 86(1997), 115–197.

[Kr] S. Kravetz. On the geometry of Teichmller spaces and the structure of their modular groups. Ann. Acad. Sci. Fenn. 278(1959), 35 pp.

[Mas1] H. Masur. On a class of geodesics in Teichmüller space. Annals of Math. 102(1975), 205–221.

[Mas2] H. Masur. The extension of the Weil-Petersson metric to the boundary of Teichmüller space. Duke Math. J. 43(1976), 623–635.

[MM1] H. Masur and Y. Minsky. Geometry of the complex of curves I: hyperbolicity. Invent. Math. 138(1999), 103–149.

[MM2] H. Masur and Y. Minsky. Geometry of the complex of curves II: hierarchical structure. Geom. & Funct. Anal. 10(2000), 902–974.

[MW] H. Masur and M. Wolf. Teichmüller space is not Gromov hyperbolic. Ann. Acad. Sci. Fenn. 20(1995), 259–267.

[MP1] J. McCarthy and A. Papadopoulos. The mapping class group and a theorem of Masur-Wolf. Topology Appl. 96(1996), 75–84.

[MP2] J. McCarthy and A. Papadopoulos. The visual sphere of Teichmller space and a theorem of Masur-Wolf. Ann. Acad. Sci. Fenn. Math. 24(1999), 147–154.

[Mc] C. McMullen. The moduli space of Riemann surfaces is Kähler hyperbolic. Annals of Math. 151(2000), 327–357.

[Min] Y. Minsky. Quasi-projections in Teichmüller space. J. Reine Angew. Math. 473(1996), 121–136.

[Wol] S. Wolpert. Noncompleteness of the Weil-Petersson metric for Teichmüller space. Pacific J. Math. 61(1975), 573–577.

Math Department, University of Chicago, 5734 S. University Ave., Chicago, IL 60637 email: brock@math.uchicago.edu, farb@math.uchicago.edu