Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 24, 1999, 147–154

THE VISUAL SPHERE OF TEICHMÜLLER SPACE AND A THEOREM OF MASUR–WOLF

John D. McCarthy and Athanase Papadopoulos

Michigan State University, Department of Mathematics East Lansing, MI 48824, U.S.A.; mccarthy@math.msu.edu C.N.R.S., Mathematiques, Université Louis Pasteur 7 rue René-Descartes, F-67084 Strasbourg Cedex, France; papadopoulos@math.u-strasbg.fr

Abstract. In [MW], Masur and Wolf proved that the Teichmüller space of genus g > 1 surfaces with the Teichmüller metric is not a Gromov hyperbolic space. In this paper, we provide an alternative proof based upon a study of the visual sphere of Teichmüller space.

1. Introduction

As observed in [MW], the Teichmüller space of surfaces of genus g > 1 with the Teichmüller metric shares many properties with spaces of negative curvature. In his study of the geometry of Teichmüller space [Kr], Kravetz claimed that Teichmüller space was negatively curved in the sense of Busemann [B]. It was not until about ten years later, that Linch [L] discovered a mistake in Kravetz’s arguments. This left open the question of whether or not Teichmüller space was negatively curved in the sense of Busemann. This question was resolved in the negative by Masur in [Ma].

A metric space X is negatively curved, in the sense of Busemann, if the distance between the endpoints of two geodesic segments from a point in X is at least twice the distance between the midpoints of these two segments. An immediate consequence of this definition is that distinct geodesic rays from a point in a Busemann negatively curved metric space must diverge. Masur proved that Teichmüller space is not negatively curved, in the sense of Busemann, by constructing distinct geodesic rays from a point in Teichmüller space which remain a bounded distance away from each other.

In [G], Gromov introduced a notion of negative curvature for metric spaces which, while less restrictive than that of Busemann, implies many of the properties which Teichmüller space shares with spaces of Riemannian negative sectional curvature. This raised the question of whether Teichmüller space was negatively curved in the sense of Gromov, (i.e. Gromov hyperbolic). According to one of the definitions of Gromov hyperbolicity, an affirmative answer to this question would rule out so-called “fat” geodesic triangles in Teichmüller space. In [MW], Masur and Wolf resolved the Gromov hyperbolicity question in the negative by constructing such “fat” geodesic triangles.

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

As observed in [MW], the existence of distinct nondivergent rays from a point in Teichmüller space does not preclude Teichmüller space from being Gromov hyperbolic. Apparently for this reason, rather than taking Masur’s construction of such rays as the starting point for their proof, Masur and Wolf found their motivation from another source. They observed that the isometry group of the Teichmüller metric is the mapping class group [R], which is not a Gromov hyperbolic group, since it contains a free abelian group of rank 2. This fact, like Masur’s result on the existence of distinct nondivergent rays from a point, is insufficient to imply that Teichmüller space is not Gromov hyperbolic. Nevertheless, it served as motivation for Masur and Wolf’s construction of “fat” geodesic triangles.

In this paper, we provide an alternative proof of the result of Masur and Wolf. Our proof, unlike that of Masur and Wolf, builds upon Masur’s construction of nondivergent rays from a point in Teichmüller space. On the other hand, unlike the proof of Masur and Wolf, our proof depends upon one of the deeper consequences of Gromov hyperbolicity. Namely, in order for Teichmüller space to be Gromov hyperbolic, the visual sphere of Teichmüller space would have to be Hausdorff. We show that, on the contrary, the visual sphere of Teichmüller space is not Hausdorff. The proof of this fact relies heavily upon the specific nature of Masur’s construction of nondivergent rays. In this way, we show that the result of Masur and Wolf that Teichmüller space is not negatively curved in the sense of Gromov is latent in Masur’s original proof that Teichmüller space is not negatively curved in the sense of Busemann.

The outline of the paper is as follows. In Section 2, we review the prerequisites for our proof. In Section 3, we prove our main result that the visual sphere of Teichmüller space is not Hausdorff and conclude that Teichmüller space is not Gromov hyperbolic.

2. Preliminaries

2.1. Teichmüller space. Let M denote a closed, connected, orientable surface of genus g ≥ 2. The Teichmüller space T_{g} of M is the space of equivalence classes of complex structures on M, where two complex structures S_{1} and S_{2} on M are equivalent if there is a conformal isomorphism h: S_{1} → S_{2} which is isotopic to the identity map of the underlying topological surface M.

The Teichmüller distance d([S_{1}], [S_{2}]) between the equivalence classes [S_{1}] and [S_{2}] of two complex structures S_{1} and S_{2} on M is defined as \frac{1}{2} log inf_{h} K(h), where the infimum is taken over all quasiconformal homeomorphisms h: S_{1} → S_{2} which are isotopic to the identity map of M and K(h) is the maximal dilatation of h.

The visual sphere of (T_{g}, d) and a theorem of Masur–Wolf 149

As shown by Kravetz [Kr], (T_{g}, d) is a straight G-space in the sense of Busemann ([B], [A]). Hence, any two distinct points, x and y, in T_{g} are joined by a unique geodesic segment (i.e. an isometric image of a Euclidean interval), [x, y], and lie on a unique geodesic line (i.e. an isometric image of R), γ(x, y).

Now, fix a conformal structure S on M and let QD(S) be the space of holomorphic quadratic differentials on S. The geodesic rays (i.e. isometric images of [0, ∞)) which emanate from the point [S] in T_{g} are described in terms of QD(S). If q is a holomorphic quadratic differential on S, p is a point on S and z is a local parameter on S defined on a neighborhood U of p, then q may be written in the form ϕ(z) dz^{2} for some holomorphic function ϕ on U. If ϕ(p) ≠ 0 and z_{0} = z(p), then on a sufficiently small neighborhood V of p contained in U, we may define a branch ϕ(z)^{1/2} of the square root of ϕ. The integral w = Φ(z) = ∫_{z_{0}}^{z} ϕ(z)^{1/2} dz is a conformal function of z and determines a local parameter for S on a sufficiently small neighborhood W of p in V . This parameter w is called a natural rectangular parameter for q at the regular point p. In terms of this parameter w, q may be written in the form dw^{2} . For each nonzero quadratic differential q on S, there is a one-parameter family {S_{K}} of conformal structures on M and quadratic differentials {q_{K}} on S_{K} obtained by replacing the natural parameters w for q on S by natural parameters w_{K} for q_{K} on S_{K}. The relationship between w_{K} and w is given by the rule:

Re w_{K} = K^{1/2} Re w, Im w_{K} = K^{−1/2} Im w.

The Teichmüller distance from [S_{K}] to [S] is equal to log(K)/2. The map t ↦ [S_{e^{2t}}] is a Teichmüller geodesic ray emanating from [S] and every geodesic ray emanating from [S] is of this form. Two nonzero quadratic differentials on S determine the same Teichmüller geodesic ray in T_{g} emanating from [S] if and only if they are positive multiples of one another.

It is well known that (T_{g}, d) is homeomorphic to R^{6g−6} and closed balls in (T_{g}, d) are homeomorphic to closed balls in R^{6g−6} . In fact, using the previous description of geodesic rays, a homeomorphism can be constructed from the open unit ball of QD(S) onto T_{g}. Suppose q is a point in the open unit ball of QD(S). Then q = kq_{1} where 0 ≤ k < 1 and q_{1} is a quadratic differential in the unit sphere of QD(S). Map q to the point [S_{K}] on the geodesic ray through [S] in the direction of q_{1} where K = (1 + k)/(1 − k). By the work of Teichmüller, this map is a homeomorphism from the open unit ball of QD(S) onto T_{g}. Since QD(S) is a complex vector space of dimension 3g − 3, this proves that T_{g} is homeomorphic to R^{6g−6} . Note also that this homeomorphism maps the closed ball of radius k centered at the origin of QD(S) onto the closed ball of radius log(K)/2 centered at the point [S] in (T_{g}, d). This proves that closed balls in (T_{g}, d) are homeomorphic to closed balls in R^{6g−6} .

We shall be particularly interested in the Jenkins–Strebel differentials. These are the quadratic differentials all of whose noncritical horizontal trajectories are closed. Let θ be a Jenkins–Strebel differential and F be the horizontal foliation of θ. The complement in M of the critical trajectories of F consists of p disjoint open annuli A_{1}, …, A_{p}, where 1 ≤ p ≤ 3g − 3. Let σ_{i} be a core curve of the annulus A_{i}. The core curves σ_{1}, …, σ_{p} are distinct, nontrivial, pairwise nonisotopic circles on M. Each annulus A_{i} is foliated by closed leaves of F isotopic to σ_{i}. Let M_{i} be the modulus of the annulus A_{i}. The basic existence and uniqueness theorem of Jenkins–Strebel ([J], [S]) states that there exists a unique quadratic differential θ in Q(S) with prescribed isotopy classes γ_{i} = [σ_{i}] of core curves and moduli M_{i} of the corresponding annuli A_{i}. Note that two Jenkins–Strebel differentials on S determine the same Teichmüller geodesic in T_{g} emanating from [S] if and only if the horizontal foliations of these Jenkins–Strebel differentials are projectively equivalent.

Following Masur [Ma], we define a Strebel ray in T_{g} emanating from [S] to be a Teichmüller geodesic ray determined by a Jenkins–Strebel differential on S. Suppose that θ_{1} and θ_{2} are Jenkins–Strebel differentials corresponding to the same isotopy classes of core curves, but not necessarily the same moduli, of corresponding annuli. Then, following Masur, we say that the Strebel rays determined by θ_{1} and θ_{2} are similar. Masur proved that similar Strebel rays emanating from the same point in T_{g} are nondivergent.

Theorem (Masur [Ma]). Let r and s be similar Strebel rays in T_{g} emanating from a point x in T_{g}. There exists N < ∞ such that if y and z are any two points on r and s which are equidistant from x, then d(y, z) ≤ N.

Since g ≥ 2, there exist distinct similar Strebel rays r and s in T_{g} emanating from the same point x = [S] in T_{g}. We may construct all such pairs of rays as follows. Choose a collection of disjoint, nontrivial, pairwise nonisotopic circles σ_{1}, …, σ_{p} on M, where 2 ≤ p ≤ 3g − 3. Let a = (a_{1}, a_{2}, …, a_{p}) and b = (b_{1}, …, b_{p}) be p-tuples of positive real numbers a_{i} and b_{i} such that a and b lie on distinct rays emanating from the origin in R^{p}. Let θ be the Jenkins–Strebel differential on S corresponding to the isotopy classes γ_{i} = [σ_{i}] of core curves and moduli a_{i} of corresponding annuli. Likewise, let ψ be the Jenkins–Strebel differential on S corresponding to the isotopy classes γ_{i} = [σ_{i}] of core curves and moduli b_{i} of corresponding annuli. Finally, let r and s be the Strebel rays determined by θ and ψ.

Combining the observation of the previous paragraph with his theorem on nondivergence of similar Strebel rays, Masur constructed distinct, nondivergent Teichmüller geodesic rays emanating from the same point in T_{g}. Indeed, any pair of distinct similar Strebel rays emanating from the same point in T_{g} is such a pair of nondivergent rays. In this way, Masur proved that T_{g} is not negatively curved in the sense of Busemann [Ma]. The particular nature of Masur’s construction of nondivergent rays will be crucial to our proof that T_{g} is not negatively curved in the sense of Gromov.

The modulus of a flat cylinder C of circumference l and height h is Mod(C) = h/l. Let S be a conformal structure on M. Every cylinder C embedded in M has a conformal structure induced from S. C is conformally equivalent to a unique flat cylinder up to change of scale. The modulus of C is the modulus of any such flat cylinder. Let γ be an isotopy class of nontrivial simple closed curves on M. The modulus mod_{S}(γ) of γ is defined to be the supremum of the moduli of all cylinders embedded in M with core curve σ ∈ γ.

For each conformal metric ϱ on S, let l_{ϱ}(γ) denote the infimum of the lengths, with respect to ϱ, of simple closed curves σ ∈ γ. Let A_{ϱ} denote the area, with respect to ϱ, of M. The extremal length ext_{S}(γ) of γ (with respect to the conformal structure S on M) is equal to sup_{ϱ}(l_{ϱ}(γ))^{2}/A_{ϱ}. The extremal length is related to the modulus by the equation ext_{S}(γ) = 1/ mod_{S}(γ).

According to Kerckhoff [K], the Teichmüller metric d may be expressed in terms of extremal length.

Theorem (Kerckhoff [K]). The Teichmüller distance between two points [S_{1}] and [S_{2}] in T_{g} is given by the rule:

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 isotopy classes γ of nontrivial simple closed curves on M.

We recall that there is a unique hyperbolic conformal metric ϱ on S. There exists a unique hyperbolic geodesic in the isotopy class γ. The hyperbolic length l_{ϱ}(γ) is the length of this hyperbolic geodesic. Maskit established the following comparisons between the hyperbolic length l_{ϱ}(γ) and the extremal length ext_{S}(γ) [M].

Theorem (Maskit ([M]). Let γ be an isotopy class of nontrivial simple closed curves on M, S be a conformal structure on M and ϱ be the unique hyperbolic conformal metric on S. Let l be the hyperbolic length l_{ϱ}(γ) and m be the extremal length ext_{S}(γ). Then l ≤ mπ and m ≤ \frac{1}{2}le^{l/2} .

2.2. Visual spheres and Gromov hyperbolicity. Let X be a space equipped with a metric d. X is said to be proper if closed balls in X are compact. Since closed balls in (T_{g}, d) are homeomorphic to closed balls in R^{6g−6} , (T_{g}, d) is proper. X is said to be geodesic if every pair of points x, y ∈ X can be connected by a geodesic segment (i.e. an isometric embedding of an interval). By Kravetz’ result that (T_{g}, d) is a straight G-space in the sense of Busemann discussed in (2.1), (T_{g}, d) is geodesic.

Let x be a point in X. A geodesic ray emanating from x is an isometric embedding r: [0, ∞) → X mapping 0 to x. If r_{1} and r_{2} are two geodesic rays in X emanating from x and the function t ↦ d(r_{1}(t), r_{2}(t)) is bounded, then we say that r_{1} and r_{2} are asymptotic and write r_{1} ∼ r_{2} . In this way, we define an equivalence relation ∼ on the set R_{x} of geodesic rays in X emanating from x. Equip R_{x} with the topology of uniform convergence on compact sets. The visual sphere of X at x is the quotient space ∂_{vis,x}X of R_{x} with respect to the equivalence relation ∼.

Gromov ([G], see also [CDP], [GH]) introduced a notion of hyperbolicity for metric spaces which is now called Gromov hyperbolicity. Gromov hyperbolic metric spaces share many of the qualitative properties of hyperbolic space. We shall not need the precise definition of Gromov hyperbolicity. We shall, however, require the following result.

Theorem (Gromov [CDP]). Let X be a proper, geodesic, Gromov hyperbolic space and x be a point in X. Then the visual sphere ∂_{vis,x}X of X at x is Hausdorff.

Remark 2.3. In fact, the visual sphere of a proper, geodesic, Gromov hyperbolic space is metrizable. The visual sphere of such a space does not depend upon the base point x in X and is naturally isomorphic to the Gromov boundary ∂X of X [CDP]. Note that the visual sphere is defined for any metric space. The Gromov boundary, however, is only defined for a restricted class of metric spaces including Gromov hyperbolic spaces.

3. The visual sphere of Teichmüller space

In this section, we prove that the visual sphere of Teichmüller space is not Hausdorff and conclude that Teichmüller space is not Gromov hyperbolic.

Theorem 3.1. Let S be a conformal structure on M representing a point x in T_{g}. Then the visual sphere ∂_{vis,x}T_{g} of T_{g} at x, with respect to the Teichmüller metric d, is not Hausdorff.

Proof. Let σ_{0} and σ_{1} be a pair of disjoint simultaneously nonseparating circles on M. For each real number t with 0 < t < 1, let θ_{t} denote the unique Jenkins–Strebel differential on S with core curves σ_{0} and σ_{1} and moduli M_{0} = 1 − t and M_{1} = t. Let θ_{0} denote the unique Jenkins–Strebel differential on S with core curve σ_{0} and modulus M_{0} = 1. Let θ_{1} denote the unique Jenkins– Strebel differential on S with core curve σ_{1} and modulus M_{1} = 1. Let r_{t} be the geodesic ray in T_{g} emanating from x corresponding to the nonzero quadratic differential θ_{t}. The family {r_{t} | 0 ≤ t ≤ 1} is a continuous one-parameter family of geodesic rays in T_{g} emanating from x. Let [r_{t}] denote the point in ∂_{vis,x}T_{g} represented by r_{t}.

Note that r_{t} is similar to r_{1/2} for all t such that 0 < t < 1. By Masur’s result on nondivergence of similar rays discussed in (2.1), it follows that r_{t} is asymptotic to r_{1/2} for all t such that 0 < t < 1. Let x = [r_{1/2}]. Then x = [r_{t}] for all t such that 0 < t < 1. By continuity of the quotient map from R_{x} to the visual sphere (recalling that the visual sphere is equipped with the quotient topology), and the convergence of the rays in R_{x}, [r_{0}] and [r_{1}] are contained in the closure of x in ∂_{vis,x}T_{g}.

We shall now show, using Maskit’s comparison of extremal and hyperbolic lengths discussed in (2.1), that [r_{0}] is not equal to [r_{1}]. Since σ_{0} and σ_{1} are simultaneously nonseparating circles on M, we may choose a nonseparating circle σ on M such that σ is disjoint from σ_{1} , transverse to σ_{0} , and meets σ_{0} in exactly one point. Let γ_{i} denote the isotopy class of σ_{i} and γ denote the isotopy class of σ. Let {S_{K}^{i}} denote the family of conformal structures on M determined by θ_{i}.

We recall Masur’s description of the surfaces {S_{K}^{i}} ([Ma]). The complement of the critical points of θ_{i} and the horizontal leaves of θ_{i} joining critical points of θ_{i} is a single annulus R^{i} foliated by closed horizontal leaves of θ_{i} homotopic to σ_{i}. We may assume that σ_{i} is the central curve of R^{i}. The surface S_{K}^{i} is formed from S by “fattening” R^{i}, by cutting M along σ_{i} and inserting a standard annulus of appropriate modulus. As K tends to infinity, the modulus of the inserted annulus tends to infinity. Hence, the modulus of γ_{i} on S_{K}^{i} tends to infinity. In other words, ext_{S_{K}^{i}} (γ_{i}) tends to zero.

In particular, ext_{S_{K}^{0}} (γ_{0}) tends to zero as K tends to infinity. Let ϱ_{K}^{0} denote the unique hyperbolic conformal metric on S_{K}^{0} . By Maskit’s comparison theorem discussed in (2.1), l_{ϱ_{K}^{0}} (γ_{0}) tends to zero as K tends to infinity. Since σ_{0} meets σ transversely and in a single point, the unique hyperbolic geodesics for the hyperbolic metric ϱ_{K}^{0} in the isotopy classes of σ_{0} and σ also meet transversely and in a single point. l_{ϱ_{K}^{0}} (γ_{0}) and l_{ϱ_{K}^{0}} (γ) are the respective lengths of these hyperbolic geodesics. Hence, by Lemma 1 of Chapter 11, Section 3.3 of [A], l_{ϱ_{K}^{0}} (γ) tends to infinity as K tends to infinity. Again, by Maskit’s comparison theorem, ext_{S_{K}^{0}} (γ) tends to infinity as K tends to infinity.

On the other hand, note that σ is disjoint from σ_{1} . Let R be any annulus on S disjoint from σ_{1} with core curve isotopic to σ. By the description of S_{k}^{1} in terms of fattening R^{1} along σ_{1} , the annulus R embeds conformally in S_{k}^{1} . Hence, the modulus of γ on S_{k}^{1} is bounded below by the constant C = mod_{S}(R). In other words, the extremal length of γ on S_{k}^{1} is bounded above by the constant 1/C.

We have shown that ext_{S_{K}^{0}} (γ) tends to infinity and ext_{S_{K}^{1}} (γ) remains bounded above as K tends to infinity. Hence, ext_{S_{K}^{0}} (γ)/ ext_{S_{K}^{1}} (γ) tends to infinity as K tends to infinity. By Kerckhoff’s description of the Teichmüller metric in terms of extremal length discussed in (2.1), d(S_{K}^{0}, S_{K}^{1}) tends to infinity as K tends to infinity. We conclude that r_{0} is not asymptotic to r_{1} . In other words, [r_{0}] ≠ [r_{1}]. Hence, we have a pair of distinct points [r_{0}] and [r_{1}] in the closure of a single point [r_{1/2}] in the visual sphere ∂_{vis,x}T_{g} of T_{g} at x. It follows that the visual sphere ∂_{vis,x}T_{g} of T_{g} at x is not Hausdorff.

154 J.D. McCarthy and A. Papadopoulos

We are now ready to deduce the result of Masur and Wolf.

Corollary 3.2 (Masur–Wolf [MW]). Teichmüller space with the Teichmüller metric is not Gromov hyperbolic.

Proof. Suppose that (T_{g}, d) is Gromov hyperbolic. Closed balls in (T_{g}, d) are compact and (T_{g}, d) is geodesic. By Gromov’s theorem on the visual sphere of a proper, geodesic, Gromov hyperbolic space discussed in (2.2), it follows that the visual sphere of Teichmüller space is Hausdorff. This contradicts Theorem 3.1. Hence, (T_{g}, d) is not Gromov hyperbolic.

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Received 4 July 1997