The fundamental inequality for cocompact Fuchsian groups

Abstract

We prove that the hitting measure is singular with respect to Lebesgue measure for any random walk on a cocompact Fuchsian group generated by translations joining opposite sides of a symmetric hyperbolic polygon. Moreover, the Hausdorff dimension of the hitting measure is strictly less than 1. A similar statement is proven for Coxeter groups. Along the way, we prove for cocompact Fuchsian groups a purely geometric inequality for geodesic lengths, strongly reminiscent of the Anderson-Canary-Culler-Shalen inequality for free Kleinian groups.

20 pages, 5 figures. Main result (Theorem 1) strengthened; results on Coxeter groups (Theorem 2) and Hausdorff dimension (Corollary 3) added. Introduction expanded