Markov chains on hyperbolic-like groups and quasi-isometries

TL;DR

This paper develops a quasi-isometry invariant theory for Markov chains on groups, focusing on hyperbolic-like groups, and proves a Central Limit Theorem for random walks on these groups.

Contribution

It introduces a new quasi-isometry invariant framework for Markov chains on hyperbolic-like groups, establishing linear progress and CLT results.

Findings

Proves linear progress for certain hyperbolic groups

Establishes a Central Limit Theorem for random walks on these groups

Demonstrates invariance under quasi-isometries for the studied properties

Abstract

We propose the study of Markov chains on groups as a "quasi-isometry invariant" theory that encompasses random walks. In particular, we focus on certain classes of groups acting on hyperbolic spaces including (non-elementary) hyperbolic and relatively hyperbolic groups, acylindrically hyperbolic 3-manifold groups, as well as fundamental groups of certain graphs of groups with edge groups of subexponential growth. For those, we prove a linear progress result and various applications, and these lead to a Central Limit Theorem for random walks on groups quasi-isometric to the ones we consider.