Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 20, 1995, 259–267

TEICHMÜLLER SPACE IS NOT GROMOV HYPERBOLIC

Howard A. Masur and Michael Wolf

University of Illinois at Chicago, Department of Mathematics Chicago, IL 60680, U.S.A.; u12341@uicvm.uic.edu Rice University, Department of Mathematics Houston, TX 77251, U.S.A.; mwolf@math.rice.edu

Abstract. We prove that the Teichmuller space of genus g > 1 surfaces with the Teichmuller metric is not a Gromov hyperbolic space.

1. Introduction

The Teichmüller space of surfaces of genus g > 1 with the Teichmüller metric is not nonpositively curved, in the sense that there are distinct geodesic rays from a point that always remain within a bounded distance of each other ([Ma1]). Despite this phenomenon, Teichmüller space and its quotient, moduli space, share many properties with spaces of negative curvature: for instance, most converging geodesic rays are asymptotic [Ma2], and the geodesic flow on the moduli space is ergodic [Ma3].

One can ask whether these properties can be explained by Teichmüller space having non-positive curvature in a sense weaker than that of Busemann used in [Ma1], which declared a space X to be negatively curved if the endpoints of two segments from p ∈ X are spread more than twice as far as the midpoints.

In his study of hyperbolic groups, Gromov ([Gr], see also [GdlH]) introduced a notion of negative curvature, now called Gromov hyperbolicity, that still captured many of the qualitative aspects of Riemannian negative sectional curvature, but was less restrictive than that of Busemann. Specifically, Gromov declared a space X to be hyperbolic if there existed a number M so that for any p ∈ X and any triangle in X with vertex at p, the leg of the triangle opposite p would be within an M -neighborhood of the legs of the triangle emanating from p. Thus, for instance, the flat Euclidean strip {(x, y) ∈ R^{2} | 0 < x < 1} would be Gromov hyperbolic but not Buseman negatively curved; moreover, the fact that there are pairs of rays emanating from p ∈ T_{g} which do not diverge does not, in itself, preclude Teichmüller space with the Teichmüller metric from being Gromov hyperbolic.

1991 Mathematics Subject Classification: Primary 32G15; Secondary 51K10.

Partially supported by the National Science Foundation, grants DMS 9201321, DMS 9300001 and DMS 9022140 (MSRI). The second author is an Alfred P. Sloan Research Fellow.

Nevertheless, the goal of this paper (Theorem 3.1) is to show that Teichmüller space is not Gromov hyperbolic. This, of course, also immediately implies that any metric quasi-isometric to the Teichmüller metric is also not Gromov hyperbolic, so any Gromov hyperbolic metric on the Teichmüller space is quite different from the Teichmüller metric.

In this connection, one needs to observe that the isometry group of the Teichmüller metric is the mapping class group ([Roy]), which contains large rank abelian subgroups, and so is not a Gromov hyperbolic group (with the word metric). This in itself does not seem to imply immediately that Teichmüller space is not Gromov hyperbolic. For example there are Kleinian groups with rank 2 abelian subgroups acting on hyperbolic 3 space, a Gromov hyperbolic space. It does suggest that good candidates for triangles to contradict Gromov’s condition might be constructed with vertices at images of a single point p under high iterates of commuting isometries.

In fact, this is the approach we take, showing (Theorem 3.1) that with respect to the Dehn twists τ_{β_{1}} and τ_{β_{2}} about disjoint curves β_{1} and β_{2} on a surface F, the triangles determined by the points x, τ_{β_{1}}^{n} · x, τ_{β_{2}}^{−n} · x contradict Gromov’s condition: the legs of this triangle are given by the Teichmüller geodesics whose corresponding Teichmüller maps from x are described explicitly in [MM], and the distances between points on the legs are estimated from below in terms of estimates of relevant extremal lengths.

We organize our discussion as follows. In Section 2, we recall the background information we will need, and set the notation. In Section 3 we state and prove our main result.

The authors are indebted to M. Kapovich for bringing this question to their attention.

2. Background and notation

2.1. Teichmüller space, metric, maps. Let M be a closed C^{∞} surface of genus g ≥ 2; everything in this note extends to punctured surfaces with only additional notation, so we concentrate on the closed surface case. We consider the Teichmüller space T_{g} with the Teichmüller metric d(·, ·). Recall that points in Teichmüller space are equivalence classes of Riemann surface structures S on M, the structure S_{1} is equivalent to the structure S_{2} if there is a homeomorphism h: M → M, homotopic to the identity, which is a conformal map of the structures S_{1} and S_{2} .

We define the Teichmüller distance d({S_{1}}, {S_{2}}) by

d(\{S_{1}\}, \{S_{2}\}) = \frac{1}{2} log inf_{h} K(h)

where h: S_{1} → S_{2} is a quasiconformal homeomorphism homotopic to the identity on M and K[h] is the maximal dilatation of h. This metric is well-defined, so we may unambiguously write S_{1} for {S_{1}}.

An extraordinary fact about this metric is that the extremal maps, known as Teichmüller maps, admit an explicit description, as does the family of maps which describe a geodesic.

Specifically, let q ∈ QD(S) denote a holomorphic quadratic differential on S. A horizontal (respectively vertical) trajectory is an arc along which q(z) dz^{2} > 0 (respectively q(z) dz^{2} < 0) except at the zeros of q. A trajectory is critical if it passes through a critical point; otherwise it is regular. If z is a local parameter near p ∈ S with q(p) ≠ 0 and z(p) = z_{0} , then w = ∫_{z_{0}}^{z} q(z)^{1/2} dz is the natural parameter q near p. The line element |q(z)|^{1/2} |dz| defines the q -metric on S.

Teichmüller’s theorem asserts that if S_{1} and S_{2} are distinct points in T_{g}, then there is a unique quasiconformal h: S_{1} → S_{2} with h homotopic to the identity on M which minimizes the maximal dilatation of all such h. The complex dilatation of h may be written μ(h) = kq̄/|q| for some non-trivial q ∈ QD(S_{1}) and some k, 0 < k < 1, and then

d(S_{1}, S_{2}) = \frac{1}{2} log(1 + k)/(1 − k).

Conversely, for each −1 < k < 1 and non-zero q ∈ QD(S_{1}), the quasiconformal homeomorphism h_{k} of S_{1} onto h_{k}(S_{1}), which has complex dilatation kq̄/|q|, is extremal in its homotopy class. Each extremal h_{k} induces a quadratic differential q′_{k} on h_{k}(S_{1}), with critical points of q and q′_{k} corresponding under h_{k}; furthermore, to the natural parameter w for q near p ∈ S_{1} there is a natural parameter w′_{k} near h_{k}(p) so that

Re w′_{k} = K^{1/2} Re w  and  Im w′_{k} = K^{−1/2} Im w,

where K = (1 + k)/(1 − k).

The map h_{k} is called the Teichmüller extremal map determined by q and k; the differential q is called the initial differential and the differential q_{k} is called the terminal differential. We can assume all quadratic differentials are normalized in the sense that

∥q∥ = ∫ |q| = 1.

The Teichmüller geodesic segment between S_{1} and S_{2} consists of all points h_{s}(S_{1}) where the h_{s} are Teichmüller maps on S_{1} determined by the quadratic differential q ∈ QD(S_{1}) corresponding to the Teichmüller map h: S_{1} → S_{2} and s ∈ [0, ∥μ(h)∥_{∞}].

The mapping class group Diff^{+}(M)/ Diff_{0}(M) acts on T_{g}. If {U_{α}, z_{α}} is an atlas defining the Riemann surface structure S, and f is a diffeomorphism of M, then f ·S is the Riemann surface structure defined by the atlas {f(U_{α}), z_{α} ∘f^{−1}}. The map f: S → f · S is then a conformal map between these two structures.

2.2. Modulus, extremal length, Jenkins–Strebel differentials, Dehn twists. The modulus of a flat cylinder C of circumference l and height h is mod(C) = h/l. For a simple closed curve γ ⊂ M, we define the modulus mod_{S}(γ) of γ to be the supremum of the moduli of all cylinders embedded in M with core curve isotopic to γ.

The extremal length ext_{S}(γ) of a curve γ on a surface M is defined to be

sup_{ϱ}(l_{ϱ}([γ]))^{2}/A_{ϱ},

where ϱ ranges over all conformal metrics on S with area 0 < A_{ϱ} < ∞ and l_{ϱ}([γ]) denotes the infimum of lengths of simple closed curves homotopic to γ. One shows that ext_{S}(γ) = 1/ mod_{S}(γ).

Kerckhoff [K] has given a characterization of the Teichmüller metric d(S_{1}, S_{2}) in terms of the extremal lengths of corresponding curves on the surfaces. He proves

(2.1)  d(S_{1}, S_{2}) = \frac{1}{2} log sup_{γ} \frac{ext_{S_{1}}(γ)}{ext_{S_{2}}(γ)}

where the supremum ranges over all simple closed curves on M.

Jenkins [J] and Strebel [Str] proved the existence of quadratic differentials q ∈ QD(S) with some prescribed trajectory topology. Specifically, they (see [Str], e.g.) showed that one could specify m disjoint simple loops γ_{1}, …, γ_{m}, with 1 ≤ m ≤ 3g − 3, on S representing distinct non-trivial free homotopy classes, and m positive numbers M_{1}, …, M_{m}, and that then one could find a unique (up to scalar multiple) quadratic differential Q = Q(z) dz^{2} ∈ QD(S) with the following property: if S′ is the result of removing the critical trajectories of Q(z) dz^{2} from S, then S′ is the union of annuli A_{1}, …, A_{m} with A_{j} homotopically equivalent to γ_{j} and the modulus of A_{j} was M_{j}, up to some fixed (independent of j) scalar multiple. Further S − S′ is the union of a finite number of analytic arcs, the smooth pieces of the critical trajectories.

Consider a point S ∈ T_{g} and consider the effect of a Dehn twist τ_{α} about a curve α ⊂ M yielding a point τ_{α}·S ∈ T_{g}. It is natural to ask for a characterization of the Teichmüller map h: S → τ_{α} · S, or more generally, for a characterization of the Teichmüller map h_{n} from S → τ_{α}^{n} · S in terms of the data α, S and n ∈ Z. This was described by Masur and Marden [MM] as follows. Let q_{α} = q_{α}(z) dz^{2} denote the Jenkins–Strebel differential determined, as above, by α ⊂ M, and suppose that α ⊂ S has modulus R. Set

m = (log R)/2π
σ_{n} = tan^{−1}(2m/n)

and

k_{n} = \frac{|n|/2m}{(1 + (n/2m)^{2})^{1/2}} .

Then [MM] the extremal map h_{n}: S → τ_{α}^{n}·S is the Teichmüller map determined by [exp(−i(σ_{n} + π))]q_{α} and k_{n}. Furthermore, if we pull back the terminal quadratic differential q′_{α} on τ_{α}^{n}·S to S via the (tautologically) conformal map τ_{α}^{n}: S → τ_{α}^{n}·S between the pullback structure S and the structure τ_{α}^{n} · S, then the pull-back differential (τ_{α}^{n})^{∗}q′_{α} satisfies

(2.2)  (τ_{α}^{n})^{∗}q′_{α} = e^{iθ}q_{α}

so that, in particular, the metrics |q_{α}| and |(τ_{α}^{n})^{∗}q′_{α}| agree.

2.3. Gromov hyperbolicity. Let X be a geodesic metric space, that is, a metric space (X, d) where every pair of points x, y ∈ X can be connected by the isometric image of the segment [0, d(x, y)]. In such a space, we can define the notion of a triangle with vertices x, y and z ∈ X to be the union of geodesic segments [xy], [yz], and [xz] connecting x and y, y and z, and x and z, respectively. Naturally, Teichmüller space with the Teichmüller metric is a geodesic metric space.

Gromov (see [GdlH]) introduced a notion of when such a space would share a number of qualitative properties with hyperbolic space, his definition now being commonly called “Gromov hyperbolicity”. We will say that

Definition 2.1. The geodesic metric space X is Gromov hyperbolic if (∗) There is a number δ ≥ 0 so that for every triangle Δ = [xy] ∪ [yz] ∪ [xz] and

every u ∈ [xy], we have d(u, [yz] ∪ [zx]) ≤ δ.

Hyperbolic space, (Riemannian) negatively curved manifolds, trees, Euclidean strips, free groups with the word metric and spheres are easily shown to be Gromov hyperbolic. On the other hand, the fundamental group of a non-compact finite volume hyperbolic n-manifold with n ≥ 3, equipped with the word metric, is not hyperbolic, because of the large rank (parabolic) abelian subgroup stabilizing a point at infinity (cusp).

3. Main theorem

The goal of this section is to prove

Theorem 3.1. Teichmüller space with the Teichmüller metric is not Gromov hyperbolic.

Proof. We consider a sequence of triangles T_{n} so that there does not exist a δ ≥ 0 with condition (∗) (in Definition 2.1) holding for all T_{n}.

All the triangles T_{n} will have a common vertex x_{0} ∈ T_{g}, chosen arbitrarily. The other vertices of the triangle T_{n} are the points y_{1} = τ_{β_{1}}^{n} ·x_{0} and y_{2} = τ_{β_{2}}^{−n} ·x_{0} , where β_{1} and β_{2} are disjoint simple closed curves on the surface M of genus g > 1.

We wish to estimate the Teichmüller distance from a point y ∈ [y_{1}y_{2}] to the other legs [x_{0}y_{1}] and [x_{0}y_{2}]. To this end, we let J_{1} dz^{2} ∈ QD(x_{0}) be the Jenkins– Strebel differential with core curves homotopic to β_{1} , and we suppose that the union of its regular trajectories determine an annulus of modulus R_{1} . We let m_{1} = (log R_{1})/2π, tan τ_{1} = 2m_{1}/n, and k_{1} = |n|(2m_{1})^{−1}(1 + (n/2m_{1})^{2})^{−1/2} , so that the Teichmüller map from x_{0} to y_{1} is determined by exp(−i(τ_{1} + π))J_{1} and k_{1} .

Let γ_{1} be a simple closed curve on M which crosses β_{1} but not β_{2} , and let γ_{2} be a simple closed curve on M which crosses β_{2} but not β_{1} . Then we claim

Lemma 3.2. For x ∈ [x_{0}y_{1}] ⊂ T_{n} ⊂ T_{g}, the extremal length, ext_{x}(γ_{2}), of γ_{2} on x is bounded independently of n.

Proof of Lemma 3.2. We begin with some more notation. Consider a quadratic differential q ∈ QD(x_{0}) and the associated singular flat Euclidean metric |q|. For a |q|-geodesic segment α, let the horizontal and vertical q -lengths of α be denoted

\begin{aligned}h_{q}(α) & = ∫_{α} | Re q^{1/2}| \\ v_{q}(α) & = ∫_{α} | Im q^{1/2}|. \end{aligned}

Then

(3.1)  |α|_{q} = (h_{q}(α)^{2} + v_{q}(α)^{2})^{1/2},

where |α|_{q} is the q -length of α. We observe that under the Teichmüller map determined by q and K with terminal quadratic differential q′, we will have the arc α remaining a q′ -geodesic arc and

(3.2)  \begin{aligned}h_{q′}(α) & = K^{1/2}h_{q}(α), \\ v_{q′}(α) & = K^{−1/2}v_{q}(α)  and \\ |α|_{q′}^{2} & = Kh_{q}(α)^{2} + K^{−1}v_{q}(α)^{2}. \end{aligned}

Of course, for fixed h_{q}(α) and v_{q}(α), equation (3.2) expresses |α|_{q′} as a convex function of K > 0.

We now specialize to the case in the statement of the lemma, where J_{1} ∈ QD(x_{0}) determines the Teichmüller geodesic arc [x_{0}y_{1}] ⊂ T_{g} and J′_{1} is the terminal differential on y_{1} . Since τ_{α}^{n}(β_{1}) = β_{1} , (2.2) implies

|β_{1}|_{J_{1}} = |β_{1}|_{J′_{1}}.

The convexity of |β_{1}| in K along [x_{0}y_{1}] forces |β_{1}|_{J_{x}} < |β_{1}|_{J_{1}} = |β_{1}|_{J′_{1}} for any of the quadratic differentials J_{x} ∈ QD(x) associated to the Teichmüller geodesic segment [x_{0}y_{1}] and any x ∈ [x_{0}y_{1}]^{0} . On the other hand, because a Teichmüller map is area preserving, this forces

(3.3)  mod_{x}(β_{1}) > mod_{x_{0}}(β_{1}) = mod_{y_{1}}(β_{1})

where mod_{x}(β_{1}) refers to the modulus of the β_{1} annulus on x ∈ [x_{0}y_{1}].

We use (3.3) in considering an alternative description of the Teichmüller map between x_{0} and x ∈ [x_{0}y_{1}]. Specifically, by the same technique of proof as that for Lemma 2.1 in [MM] (see also the statement for the annulus in [MM; §1.3]), we can represent the Teichmüller map between x_{0} and x ∈ [x_{0}y_{1}] as T_{α} ∘ S_{a} where T_{α} is a “partial” Dehn twist of the initial Jenkins–Strebel annulus by an angle 2πα and S_{a} is a radial expansion or (possibly) contraction of that annulus: we observe however that by (3.3), the map S_{a} is always an expansion.

Thus, we can build a model of any terminal Jenkins–Strebel differential J_{x} ∈ QD(x) with x ∈ [x_{0}y_{1}] as given by an operation of conformal plumbing followed by a partial Dehn twist, as follows. We cut the conformal cylinder along a core curve. We then glue in one cylinder to each edge of the cut, again leaving a pair of boundary components. Finally, we glue these free edges together after twisting by some angle.

The homotopy class of γ_{2} is represented by a union of geodesic segments on the boundary of the Jenkins–Strebel annulus for J_{1} . Therefore, we can find an annulus A_{2} , embedded around γ_{2} , and also disjoint from the core curve along which our initial cut (of the previous paragraph) is made. That annulus A_{2} will be unaffected by the plumbing and twisting, and so we can conclude that for all x ∈ [x_{0}y_{1}] for which x = T_{a} ∘ S_{α}x_{0} , we can find an embedded annulus A_{2} about γ_{2} of modulus bounded uniformly away from zero, independently of n.

Thus the extremal length of γ_{2} is then uniformly bounded above, independently of n, concluding the proof of the lemma.

Remark. The lemma of course holds with γ_{1} and [x_{0}y_{2}] in place of γ_{2} and [x_{0}y_{1}], by an interchange of notation in the proof.

Conclusion of the proof of Theorem 3.1. Now consider the Teichmüller geodesic arc [y_{1}y_{2}]. The Teichmüller map from y_{1} to y_{2} is given by taking a negative twist n times about β_{1} and about β_{2} . Consider the Strebel differential Q ∈ QD(y_{1}) of two annuli with core curves homotopic to β_{1} and β_{2} , of equal moduli R (see [Str]). Let m, σ_{n} and k_{n} be as in Section 2.2; then the Teichmüller map from y_{1} to y_{2} is determined by exp(−i(σ_{n}+π))Q and k_{n}. Let Q′ be the terminal differential on y_{2} .

By Lemma 3.2 and the fact that Q is a competing metric in the definition of extremal length, we have

(3.4)  |v_{Q}(γ_{2})| ≤ |γ_{2}|_{Q} ≤ ext_{y_{1}}(γ_{2})^{1/2} = O(1).

Since y_{2} = τ_{β_{2}}^{−n} · x_{0} , we have

ext_{y_{2}}(γ_{2}) = ext_{x_{0}}(τ_{β_{2}}^{n}(γ_{2})).

Since τ_{β_{2}}(γ_{2}) crosses β_{2} n times, there is a constant c_{0} > 0 so that

ext_{y_{2}}(γ_{2}) ≥ c_{0}n^{2}.

Moreover, since we can always compare any two normalized metrics on the fixed surface y_{2} , conformally equivalent to x_{0} , we find that

(3.5)  |γ_{2}|_{Q′} ≥ cn

for some c > 0.

Next, since

k_{n} = (1 + (\frac{log R}{π|n|})^{2})^{−1/2}

we see that

(3.6)  K_{n} = \frac{1 + k_{n}}{1 − k_{n}} ≍ n^{2}

where a ≍ b if their ratio is bounded above and below away from 0. Then, applying (3.4), (3.5) and (3.6) to the identity

K_{n}h_{Q}(γ_{2})^{2} + K_{n}^{−1}v_{Q}(γ_{2})^{2} = |γ_{2}|_{Q′}^{2}

yields

(3.7)  h_{Q}(γ_{2}) > c_{2} > 0.

Next, we observe that −Q is the terminal quadratic differential on y_{1} for the Teichmüller map from y_{2} to y_{1} , with initial differential −Q′. Then the same argument as above shows that h_{−Q′}(γ_{1}) > c′_{2} > 0, independently of n. We can then apply formula (3.2) again to conclude that h_{−Q}(γ_{1}) > c_{3}n for some c_{3} > 0, which, of course, is equivalent to

(3.8)  v_{Q}(γ_{1}) > c_{3}n.

Finally, consider the point y_{∗} ∈ [y_{1}y_{2}] determined by the Teichmüller map defined by Q with K^{1/2} = \sqrt{n}; let Q_{∗} ∈ QD(y_{∗}) denote the terminal differential. Then (3.7) and (3.8), along with the relationship (3.2) show that

\begin{aligned}|γ_{2}|_{Q_{∗}} & ≥ h_{Q_{∗}}(γ_{2}) ≥ c_{2}\sqrt{n} and \\ |γ_{1}|_{Q_{∗}} & ≥ v_{Q_{∗}}(γ_{1}) ≥ c_{3}\sqrt{n}. \end{aligned}

Since Q_{∗} is a competing metric for extremal length, ext_{y_{∗}}(γ_{i}) ≥ |γ_{i}|_{Q_{∗}}^{2} > c_{4}n.

Finally, we apply Kerckhoff’s formula (2.1) and Lemma 3.2 to estimate the Teichmüller distance d([x_{0}y_{1}], y_{∗}): we see that since ext_{y_{∗}}(γ_{2}) > c_{4}n while ext_{x}(γ_{2}) < c_{5} for x ∈ [x_{0}y_{1}], then (2.1) forces d(x, y_{∗}) > \frac{1}{2} log(c_{5}^{−1}c_{4}n). Since an analogous estimate holds for d([x_{0}y_{2}], y_{∗}), we see that the defining condition (∗) of Definition 2.1 of Gromov hyperbolicity does not hold.

Teichmüller space is not Gromov hyperbolic 267

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Received 4 November 1993