The fundamental inequality for cocompact Fuchsian groups
Dynamical Systems
2021-04-28 v2 Geometric Topology
Probability
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.
Cite
@article{arxiv.2012.07417,
title = {The fundamental inequality for cocompact Fuchsian groups},
author = {Petr Kosenko and Giulio Tiozzo},
journal= {arXiv preprint arXiv:2012.07417},
year = {2021}
}
Comments
20 pages, 5 figures. Main result (Theorem 1) strengthened; results on Coxeter groups (Theorem 2) and Hausdorff dimension (Corollary 3) added. Introduction expanded