The fundamental inequality for cocompact Fuchsian groups

TL;DR

This paper demonstrates that for certain hyperbolic groups, the hitting measure of random walks is singular and has Hausdorff dimension less than one, revealing deep geometric and measure-theoretic properties.

Contribution

It establishes the singularity and dimension bounds of hitting measures for cocompact Fuchsian groups and related Coxeter groups, introducing a new geometric inequality for geodesic lengths.

Findings

Hitting measure is singular with respect to Lebesgue measure.

Hausdorff dimension of the hitting measure is less than 1.

A new geometric inequality for geodesic lengths is proven.

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.