Entropy and drift for word metric on relatively hyperbolic groups

TL;DR

This paper investigates the relationship between entropy, drift, and Green distance in random walks on relatively hyperbolic groups, extending known results from hyperbolic groups and identifying conditions for equality or strict inequality.

Contribution

It generalizes the Guivarc'h inequality analysis to relatively hyperbolic groups and characterizes when the Green distance aligns with the word distance.

Findings

Equality in Guivarc'h inequality implies Green and word distances are roughly similar.

For certain relatively hyperbolic groups, the inequality is always strict.

Results extend hyperbolic group properties to a broader class of groups.

Abstract

We are interested in the Guivarc'h inequality for admissible random walks on finitely generated relatively hyperbolic groups, endowed with a word metric. We show that for random walks with finite super-exponential moment, if this inequality is an equality, then the Green distance is roughly similar to the word distance, generalizing results of Blach{\`e}re, Ha{\"i}ssinsky and Mathieu for hyperbolic groups [4]. Our main application is for relatively hyperbolic groups with respect to virtually abelian subgroups of rank at least 2. We show that for such groups, the Guivarc'h inequality with respect to a word distance and a finitely supported random walk is always strict.