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Abstract. We prove that for the surface defined by a holomorphic quadratic differential, the set of directions such that the corresponding Teichmuller geodesic lies in a compact set in the corresponding stratum is a winning set in Schmidt game. This generalizes a classical result in the case of the torus due to Schmidt and strengthens a result of Kleinbock and Weiss.

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In this paper we explore the idea that Teichmuller space with the Teichmuller metric is hyperbolic ``on average." We consider several different measures on Teichm\"uller space and show that with respect to each one, the average distance between points in a ball of radius r is asymptotic to 2r, which is as large as possible.

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We give a distance estimate for the metric on the disk complex and show that it is Gromov hyperbolic. As another application of our techniques, we find an algorithm which computes the Hempel distance of a Heegaard splitting, up to an error depending only on the genus.

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In this paper we prove that the Weil-Petersson geodesic flow is ergodic on moduli space

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We show that the subsurface projection of a train track splitting sequence is an unparameterized quasi-geodesic in the curve complex of the subsurface. For the proof we introduce induced tracks, efficient position, and wide curves. This result is an important step in the proof that the disk complex is Gromov hyperbolic. As another application we show that train track sliding and splitting sequences give quasi-geodesics in the train track graph, generalizing a result of Hamenstadt

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We use ending laminations for Weil-Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for Weil- Petersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equivalent condition for Weil-Petersson geodesics. As an application, we show the Weil-Petersson geodesic flow has compact invariant subsets with arbitrarily large topological entropy.

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This chapter is a survey of recent results in Teichmuller geometry

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We consider billiards in a certain rectangle with a horizontal barrier This gives a one parameter family of flows in different directions. We study the Hausdorff dimension of the set of directions such that the flow in that direction is not ergodic. The dimension is computed explicitly in terms of the continued fraction expansion of the length of the barrier.

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We construct a metric simplicial complex which is an almost isometric model of the moduli space M(S) of Riemann surfaces. We then use this model to compute the "tangent cone at infinity" of M(S): it is the topological cone on the quotient of the complex of curves C(S) by the mapping class group of S, endowed with an explicitly described metric. The main ingredient is Minsky's product regions theorem.

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We study the asymptotic geometry of Teichm\"uller geodesic rays. The question of whether two rays through a given point stay bounded distance apart or not was settled except for one outstanding case. In this paper we settle that last case. We show that when the transverse measures to the vertical foliations of the quadratic differentials determining two different rays are topologically equivalent, but are not absolutely continuous with respect to each other, then the rays diverge in Teichm\"uller space.

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Let S be a closed, oriented surface with a finite (possibly empty) set of points removed. In this paper we relate two important but disparate topics in the study of the moduli space M(S) of Riemann surfaces: Teichm\"{u}ller geometry and the Deligne-Mumford compactification. We reconstruct the Deligne-Mumford compactification (as a metric stratified space) purely from the intrinsic metric geometry of M(S) endowed with the Teichm\"{u}ller metric. We do this by first classifying (globally) geodesic rays in M(S) and determining precisely how pairs of rays asymptote. We construct an ``iterated EDM ray space'' functor, which is defined on a quite general class of metric spaces. We then prove that this functor applied to M(S) produces the Deligne-Mumford compactification.

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We define an ending lamination for a Weil-Petersson geodesic ray. Despite the lack of a natural visual boundary for the Weil-Petersson metric, these ending laminations provide an effective boundary theory that encodes much of its asymptotic CAT(0) geometry. In particular, we prove an ending lamination theorem for the full-measure set of rays that recur to the thick part, and we show that the association of an ending lamination embeds asymptote classes of recurrent rays into the Gromov-boundary of the curve complex. As an application, we establish fundamentals of the topological dynamics of the Weil-Petersson geodesic flow, showing density of closed orbits and topological transitivity.

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We analyze the coarse geometry of the Weil-Petersson metric on Teichm\"uller space, focusing on applications to its synthetic geometry (in particular the behavior of geodesics). We settle the question of the strong relative hyperbolicity of the Weil-Petersson metric via consideration of its coarse quasi-isometric model, the "pants graph." We show that in dimension~3 the pants graph is strongly relatively hyperbolic with respect to naturally defined product regions and show any quasi-flat lies a bounded distance from a single product. For all higher dimensions there is no non-trivial collection of subsets with respect to which it strongly relatively

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Hubert-Schmidt and McMullen have found examples of translation surfaces whose Veech group is infinitely generated. In this paper we show first that the Hubert-Schmidt examples satisfy the topological dichotomy property that for every direction either the flow in that direction is completely periodic or minimal. More significantly we show that they have minimal but non uniquely ergodic directions.

This is a list of open problems in the subject and is available as a pdf file.

This survey is available as a pdf file.

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The paper is available as a Postscript file.

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Asymptotic formulas on flat surfaces

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In a certain sense this hyperbolicity is an explanation of why the Teichmuller space has some negative-curvature properties in spite of not being itself hyperbolic: Hyperbolicity in the Teichmuller space fails most obviously in the regions corresponding to surfaces where some curve is extremely short. The complex of curves exactly encodes the intersection patterns of this family of regions (it is the "nerve" of the family), and we show that its hyperbolicity means that the Teichmuller space is "relatively hyperbolic" with respect to this family. A similar relative hyperbolicity result is proved for the mapping class group of a surface.

The paper is available as a Postscript file (563K), or (without the figures) as a dvi file (180K).

The paper (November 2000) is available as a Postscript file (1180K).