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In topology, geometry and complex analysis, one attaches a number of interesting mathematical objects to a surface S. The TeichmÃ¼ller space T (S) is the parameter space of conformal (or hyperbolic) structures on S, up to isomorphism isotopic to the identity. The Mapping Class Group Mod(S) is the group of auto-homeomorphisms of S, up to isotopy. Theâ€¦ (More)

In this paper we continue our geometric study of Harveyâ€™s Complex of Curves [12], a finite dimensional and locally infinite complex C(S) associated to a surface S, which admits an action by the mapping class group Mod(S). The geometry and combinatorics of C(S) can be applied to study grouptheoretic properties of Mod(S), and the geometry of Kleinianâ€¦ (More)

This paper concerns the interplay between the complex structure of a Riemann surface and the essentially Euclidean geometry induced by a quadratic differential. One aspect of this geometry is the " trajectory structure" of a quadratic differential which has long played a central role in Teichmfiller theory starting with Teichmiiller's proof of the existenceâ€¦ (More)

A theory of random walks on the mapping class group and its non-elementary subgroups is developed. We prove convergence of sample paths in the Thurston compact-iication and show that the space of projective measured foliations with the corresponding harmonic measure can be identiied with the Poisson boundary of random walks. The methods are based on anâ€¦ (More)

To every compact orientable surface one can associate following Harvey Ha Ha a combinatorial object the so called complex of curves which is analogous to Tits buildings associated to semisimple Lie groups The basic result of the present paper is an analogue of a fundamental theorem of Tits for these complexes It asserts that every automorphism of theâ€¦ (More)

- Howard A. Masur
- 1995

Let S be a surface of genus g with n punctures, and assume 3g âˆ’ 3 + n > 1. Associated to S is an object C(S) called the complex of curves, whose vertices are homotopy classes of nontrivial, nonperipheral simple closed curves. A k-simplex of C(S) is a collection of k + 1 disjoint nonperipheral homotopically distinct simple closed curves. The dimension of theâ€¦ (More)

Let F = Fg,n be a surface of genus g with n punctures. We assume 3g âˆ’ 3 + n > 1 and that (g, n) 6= (1, 2). The purpose of this paper is to prove, for the Weil-Petersson metric on Teichmuller space Tg,n, the analogue of Roydenâ€™s famous result [15] that every complex analytic isometry of Tg,0 with respect to the Teichmuller metric is induced by an element ofâ€¦ (More)

1 Polygonal billiards, rational billiards 3 1.1 Polygonal billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Examples: a pair of elastic point-masses on a segment and a triple of point-masses on a circle . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Unfolding billiard trajectories, rational polygons . . . . . . . . . . . . 5â€¦ (More)

We analyze the coarse geometry of the Weil-Petersson metric on TeichmÃ¼ller 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 inâ€¦ (More)

In this paper we consider billiards in a square with a barrier. We use Ratnerâ€™s theorem to compute the asymptotics for the number of closed orbits.