Fundamental inequality for hyperbolic Coxeter and Fuchsian groups equipped with geometric distances

TL;DR

This paper demonstrates that for many hyperbolic reflection and Fuchsian groups, the hitting measure of certain random walks differs fundamentally from the Lebesgue measure, revealing new insights into their geometric and probabilistic properties.

Contribution

It establishes a fundamental inequality showing the non-equivalence of hitting and Lebesgue measures for a broad class of random walks on these groups.

Findings

Hitting measure is not equivalent to Lebesgue measure for many hyperbolic groups.

The result applies to a large class of nearest-neighbour random walks.

Provides new inequalities relating geometric distances and measures.

Abstract

We prove that the hitting measure is not equivalent to the Lebesgue measure for a large class of nearest-neighbour random walks on hyperbolic reflection groups and Fuchsian groups.