Steven P. Kerckhoff

Ergodicity of billiard flows and quadratic differentials

On considere un systeme billard de 2 objets de masses m 1 et m 2 . On montre que pour un ensemble dense de paires (m 1 ,m 2 ) ce systeme est ergodique

The asymptotic geometry of teichmuller space

INTRODUCTION TEICHMULLER SPACE is the space of conformal structures on a topological surface MR of genus g where two are equivalent if there is a conformal map between them which is homotopic to the identity. This space will be denoted by Tg. Teichmuller proved that when g 12 Tg is homeomorphic to an open 6g-6 dimensional ball. Moreover, his proof showed that this homeomorphism could be realized by the radial map along geodesic rays from a fixed base point. In particular, this homeomorphism gives a natural way to compactify TR by putting the endpoints on the rays. We denote the resulting closed 6g-6 dimensional ball by Fg. The immediate question to ask is to what extent i’g depends on the base point from which Tg was compactified. In this paper we show that the geometry along certain rays depends strongly on their base points. Any diffeomorphism of MB induces an isometry of T,. The group of isometries of Tg induced by the group of diffeomorphisms of MB is called the modular group and is denoted by Mod(g). Since F* is defined in terms of geodesics, a natural question to pose is whether or not the action of Mod (g) extends continuously to Tr There are several reasons to be interested in this questions. First, a continuous map on a closed ball is easier to understand than one on an open ball since it always has a fixed point. This fact has been successfully used by Thurston who compactified Tg in such a way that the action of Mod (g) extended continuously. By examining the fixed points of elements of Mod(g), he gave a geometric description of a canonical element of each connected component of Diff M. Thurston’s compactification, TgT, is also homeomorphic to a closed 6g-6 dimensional ball so it is reasonable to ask if his and Teichmuller’s compactifications are the same; i.e., whether the identity map on the interiors extends to a homeomorphism from ?=g to TgT. There was some evidence that this was true. (See [5] and Theorem 3 below.) However we show (Theorem 2) that the compactifications are distinct. Secondly, compactification by geodesic rays have been used extensively by Mostow (and by numerous others) to study complete hyperbolic manifolds. The covering translations of such a manifold, acting on its universal cover, hyperbolic n-space, H”, extend to the closure R”. The boundary sphere of 8” (the “sphere at infinity”) is naturally identified with the space of rays through any interior point of H”. By studying the action of the fundamental group on the sphere at infinity, Mostow proved his well-known rigidity theorem. The allusion to hyperbolic manifolds is not pure whimsy; the Teichmuller space for the torus is isometric to HZ and Mod.(g) (which is isomorphic to SL(2, Z)) extends continuously to its closure. Moreover, T, was thought to have negative curvature for several years. However, Linch[8] found a mistake in the proof and Masur[9] later showed that T, is, in fact, not negatively curved. Thus the question of the extension of Mod (g) to Tg can be thought of both as a question of generalizing a result which is

The Nielsen realization problem

Closed, oriented surfaces of genus g > 2 admit many hyperbolic (constant Gaussian curvature -1) metrics in contrast to Mostows rigidity theorems in higher dimensions. Only special hyperbolic surfaces have non-trivial groups of isometries, but many different, non-isomorphic groups arise for different symmetric metrics. The group of isometries of a closed hyperbolic surface M2 is always finite and the only isometry isotopic to the identity is the identity itself. As a result, hyperbolic surfaces with non-trivial groups of isometries have been a primary source for the construction of finite subgroups of the group of isotopy classes of diffeomorphisms of M2, ?TDiff(M2). An old question, usually referred to as the Nielsen Realization Problem, is whether every such finite subgroup arises as a group of isometries of some hyperbolic surface. In this paper we answer the question in the affirmative.

Local rigidity of hyperbolic manifolds with geodesic boundary

Let W be a compact hyperbolic n-manifold with totally geodesic boundary. We prove that if n>3 then the holonomy representation of pi_1 (W) into the isometry group of hyperbolic n-space is infinitesimally rigid.

From the hyperbolic 24-cell to the cuboctahedron

We describe a family of 4-dimensional hyperbolic orbifolds, constructed by deforming an infinite volume orbifold obtained from the ideal, hyperbolic 24-cell by removing two walls. This family provides an infinite number of infinitesimally rigid, infinite covolume, geometrically finite discrete subgroups of the isometry group of hyperbolic 4-space. It also leads to finite covolume Coxeter groups which are the homomorphic image of the group of reflections in the hyperbolic 24-cell. The examples are constructed very explicitly, both from an algebraic and a geometric point of view. The method used can be viewed as a 4-dimensional, but infinite volume, analog of 3-dimensional hyperbolic Dehn filling.