Thin Triangles and a Multiplicative Ergodic Theorem for Teichmüller Geometry

1.1. Overview. In this paper, we prove a curvature-type result about Teichmüller space, in the style of synthetic geometry. We show that, in the Teichmüller metric, “thin-framed triangles are thin”—that is, under suitable hypotheses, the variation of geodesics obeys a hyperbolic-like inequality. This theorem has applications to the study of random walks on Teichmüller space. In particular, an application is worked out for the action of the mapping class group: we show that geodesics track random walks sublinearly. Recall that the Teichmüller space Tg,n is a parameter space for marked metrics on oriented surfaces of a fixed topological type (the marking is a choice of generators for π1; the type (g, n) is the genus and number of punctures or boundary components, chosen so that Σg,n is a hyperbolic surface). The points of Tg,n are conformal classes of marked metrics—equivalently, since there is exactly one metric of constant curvature in each conformal class, each point can be identified with a (marked) Poincaré metric on the surface Σg,n. The mapping class group Mod(g, n) is the collection of isotopy classes of orientation-preserving diffeomorphisms of Σg,n. For a small value ǫ, the cusps of Tg,n are the regions containing metrics on Σg,n with some nontrivial curve shorter than ǫ; the complement of the cusps is called the thick part. For any two points of Teichmüller space, there are quasiconformal maps between them; Teichmüller showed that there is a unique quasiconformal map of minimal dilatation (the eccentricity of its ellipse field) [33]. He defined a distance function accordingly, though it yields only a Finsler—not a Riemannian—metric. This Teichmüller metric is one of several natural metrics on Tg,n. Here, we will restrict attention to this choice of metric, and to Teichmüller spaces Tg of compact hyperbolic surfaces. There is a long and involved history of studying the metric geometry of Teichmüller space since the introduction of the Teichmüller metric in the late 1930s. In 1959, Kravetz [17] had a much-cited result purporting to show that the Teichmüller metric was Busemann non-positively curved (that is, that it had a convex distance function). Linch showed that this argument was incorrect in her thesis of 1971 [18], and Masur proved that Teichmüller space was in fact not Busemann non-positively curved and not CAT(0) (a slightly stronger condition) in his own thesis of 1975 [19].