Gromov Hyperbolicity , Teichmüller Space and Bers Boundary

We present in this paper a new proof of a theorem by Wolf-Masur stipulating that Teichmüller space of surface with genus g ≥ 2 equipped with the Teichmüller metric is not hyperbolic in the sense of Gromov, by constructing a family of points that converge to the Bers boundary contradicting a property proved by Bers in 1983. To our knowledge, there are several different proofs of this result, besides the original of Masur-Wolf (1975) available in the literature, see MacCarthy-Papadopoulos (1999a, 1999b), and Ivanov (2001).


Introduction
The notion of negative curvature of Teichmüller space has a long history.It starts in the late 50's of the last century with Kravetz (1959), who claimed that the Teichmüller space was negatively curved in the Busemann sense.It was thought so, until Linch exhibited in her Columbia thesis a flaw in the Kravetz's argument, reopening the question of negative curvature of the Teichmüller space.Masur, in 1975, answered in the negative this old-new question, by constructing two geodesic rays emanating from the same point staying at a bounded distance apart.Recently, Gromov in 1987, introduced his revolutionary notion of hyperbolicity for groups and more generally, for metric spaces.It is well known that even with this less restrictive notion of negative curvature, the Teichmüller space is not Gromov hyperbolic (Masur-Wolf Theorem).In this paper, we will present a new proof (by contradiction) of the Masur-Wolf Theorem by constructing a family of points that converges to the Bers boundary that contradicts a result proved by Bers in 1983, if we assume that Teichmüller space is Gromov hyperbolic.
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 (Masur-Wolf Theorem).

Teichmüller Space, Metric
Let M be a closed, connected, orientable surface of genus g ≥ 2; we consider the Teichmüller space T g with the Teichüller metric d(•, •).The points in T g are equivalence classes of conformal (complex) structures on M, where two conformal structures S 1 and S 2 on M are declared equivalent if there is a conformal homeomorphism h: S 1 → S 2 which is homotopic to the identity map of the underlying topological surface M. The Teichmüller distance is defined as d(S 1 , S 2 ) = 1 2 log inf K( f ) where the infimum is taken over all quasiconformal homeomorphisms f : S 1 → S 2 which are homotopic to the identity on M and K( f ) is the maximal dilatation of f .
An amazing fact about the extremal maps, known as Teichmüller map, that they admit an explicit description, as does the family of maps which describe a geodesic (isometric image of R).
This description is expressed in terms of quadratic differentials.Let q ∈ QD(S 1 ) denote a holomorphic quadratic differential on S 1 .If z is a local parameter near p ∈ S 1 with q(p) 0 and z(p) = z 0 , then w = z z 0 (q(z)) 1 2 dz is the natural parameter of q near the point p.
Teichmüller's theorem asserts that if S 1 , S 2 are distinct points in T g , then there is unique quasiconformal h: S 1 → S 2 with h isotopic to the identity which minimizes the maximal dilatation of all such h.The complex dilatation of h may be written μ(h) = k q/|q| for some non trivial quadratic q ∈ QD(S 1 ) and some k, 0 < k < 1, and then Conversely, for each |k| < 1 and a non-zero q ∈ QD(S 1 ), the quasiconformal homeomorphism h k of S 1 onto h k (S 1 ), with complex dilatation k q/|q|, is extremal in its isotopy class.Each extremal map h k induces a quadratic differential q k on h k (S 1 ) so that The map h k is called the Teichmüller extremal map determined by q and k.
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: We recall now a very well known result, that we will use in the proof of the main result.According to the Uniformization Theorem, each point x in Techmüller space T g can be represented as the quotient of the upper half plane H 2 by a Fuchsian group G (i.e., a discrete subgroup of PS L(2, R).) Therefore we can write x = H 2 /G.Since we assumed the topological surface M compact, then any element A of the Fuchsian group G is hyperbolic.
(i.e., trace(A) 2 > 4.) If we denote by π the natural projection H 2 → H 2 /G then the projection of the axis of the hyperbolic element A (i.e., a geodesic in H 2 invariant by A) is closed geodesic in x ∈ T g .We have the following useful relation between the trace of A and the hyperbolic length of the closed geodesic α (i.e., l h (α)).Needless to say the metric used in the measure of the length of α is nothing but, the unique hyperbolic metric h in the conformal structure x.We have: Proposition 1 Let x be a conformal structure defined on the underlying topological surface M, and h be the unique hyperbolic structure on x.Then For a proof, the reader can consult Fathi, Laudenbach, and Poénaru (1979, Lemma 1, p. 135).

Modulus, Extremal Length
The modulus of a flat cylinder C of circumference l and height h is For a simple closed curve γ ⊂ S , we define the modulus Mod S (γ) of γ to be the supremum of the moduli of all cylinders embedded in S with core curve isotopic to γ.
The extremal length ext S 0 (γ) of a curve γ on a surface S 0 is defined to be where ρ ranges over all conformal metrics on S 0 with area A ρ satisfying 0 < A ρ < ∞, and where l ρ ([γ]) denotes the infimum of lengths of simple closed curves homotopic to γ.One can show that

Maskit's Estimates
Maskit (1985) has compared the extremal and hyperbolic lengths of closed curves on any compact orientable surface M with genus g ≥ 2.
Theorem 1 Let x be a conformal structure defined on the underlying topological surface M, and h be the unique hyperbolic structure on x.Then l h (γ) ≤ πext x (γ) (4)

Extremal Quasiconformal Map in the Homotopy Class of Dehn Twist
Jenkins (1957) and Strebel (1984) proved the existence of quadratic differentials q ∈ QD(S ) with some topological conditions on the trajectories.More precisely, they proved that one could choose p disjoint simple closed curves γ 1 . . .γ p with 1 ≤ p ≤ 3g−3, on the surface M representing an admissible system of curves, and p positive numbers m 1 . . .m p , and one could find a unique (up to scalar multiplication) quadratic differentials Q = Q(z)dz 2 ∈ QD(S ) with the following property: if S is the surface after removing the critical trajectories of Q(z)dz 2 , then S is the union of annuli A 1 . . .A p with A j homotopically equivalent to γ j and the modulus of the annulus A j is M j , up to some fixed (independent of j) scalar multiple.Further, S − S is the union of finite number of analytic arcs, the smooth pieces of the critical trajectories.
The mapping class group Γ g of M is the group of isotopy classes of orientation preserving homeomorphism M → M. Γ g acts on T g by pulling back conformal structures S on M. It follows that the action of Γ g on T g is by isometries.It is a well known fact that this action is properly discontinuous on T g .
Fix an arbitrarily point S ∈ T g and consider the effect of Dehn twists τ α 1 ; about the curve α 1 , on M. It is legitimate to characterize the Teichmüller map h: S → τ α 1 (S ), in terms of: τ α 1 , S and n ∈ Z.Let q [α 1 ] denote the Jenkins-Strebel differential determined as above and suppose that α 1 ⊂ S has modulus R. Set Marden-Masur in 1975 gave the following description of the extremal map h n : S → τ α 1 • S is the Teichmüller map determined by exp (−i(σ n + π)) • q α 1 and the multiplier k n .

Gromov Hyperbolicity
A geodesic metric space (X, d) is a metric space where every couple of points x, y ∈ X can be connected by the isometric image of the segment [0, d(x, y)], we call such path geodesic segment and we denote it by [x, y].In such space, it is natural to define the notion of a triangle having any three points x, y, and z ∈ X as vertices, to be the

Main Result
The purpose of this section is to present a proof of the following result (Masur-Wolf Theorem): Main Theorem The Teichmüller space of a hyperbolic surface equipped with the Teichmüller metric is not Gromov hyperbolic.
Proof of the main theorem.We consider a sequence of triangles T n , having a common vertex x 0 ∈ T g , chosen arbitrarily.The other vertices of the triangle T n are the points y 2n = τ 2n α 1 (x 0 ) and z 2n = τ −2n α 2 (x 0 ), where α 1 and α 2 are disjoint simple closed curves on the surface M of genus g ≥ 2.
We denote by w n the midpoint of the geodesic segment [y 2n , z 2n ]; and by y n (respectively z n ) the point on the geodesic segment [x 0 , y 2n ], (respectively [x 0 , z 2n ]) such that Now if we assume that the Teichmüller space is hyperbolic then we have: We have the following claim Lemma 1 If we assume that d(w n , y n ) ≤ δ, then the sequence (y n ) ⊂ T g does not stay in any compact subset of T g .
Proof of Lemma 1.Using the triangle inequality we have we may easily conclude that, By construction of the point Using formula 7, we obtain Combining formula (1) and letting n go to ∞, we may conclude that in the other hand, we have: therefore, using formula (9), we may conclude that d(x 0 , y n ) becomes very large whenever the order of the Dehn twist n becomes in its turn large too.Which means that the sequence (y n ) does not stay in any compact subset of the Teichmüller space T g .

Remark
The previous lemma holds for (z n ) if we assume that the second inequality in ( 8) is true, and by interchanging the notations.

Conclusion of the proof of the main Theorem.
Consider now, an alternative description of the Teichmüller map from x 0 to y n , respectively from x 0 to z n , by the same techniques of proof as that of Lemma 2.1 in Marden and Masur (1975), we can represent the Teichmüller map between x 0 to y n , (respectively x 0 to z n ) as τ θ • T a where τ θ is Dehn twist of the initial Jenkins-Strebel annulus A α 1 , (respectively A α 2 ), having α 1 , (respectively α 2 ), as core curves by an angle 2π • θ and T a is a radial expansion or possibly contraction of these annuli, but we can see that in fact T a is an expansion by adopting the same technique to establish the inequality (3.3) p. 265 in Masur and Wolf (1995) for each annulus.The modulus of α 1 , (respectively α 2 ) is increasing indefinitely along the geodesic segment connecting x 0 to y n (respectively x 0 to z n ).Therefore, by the formula (3), the extremal length of α 1 , (respectively α 2 ,) is decreasing indefinitely, along the geodesic segment connecting x 0 to y n (respectively x 0 to z n ).By the Maskit's inequality (7), we may conclude that the hyperbolic length l y n (α 1 ), (respectively l z n (α 2 ),) becomes arbitrarily small whenever n becomes arbitrarily large.Therefore, according to the equality (2), the square of the trace of the hyperbolic element A 1 ∈ G y n (respectively A 2 ∈ G z n ) belonging to the Fuchsian group G w n (respectively G z n ), that uniformize the Riemann surface y n , (respectively z n ) covering the closed geodesic freely homotopic to α 1 over y n (respectively α 2 over z n ) has limit 4 when n goes to infinity.Therefore G y n and G z n converge to B-groups G y ∞ and G z ∞ respectively in the Bers boundary ∂T g of T g , each of them contains one and only one accidental parabolic transformation χ y ∞ (A 1 ) (respectively χ z ∞ (A 2 )).