The CLT for lamplighter groups with an acylindrically hyperbolic base

TL;DR

This paper establishes a Central Limit Theorem for the drift of random walks on lamplighter groups with acylindrically hyperbolic bases, extending previous results to more general group settings.

Contribution

It proves a CLT for the drift of random walks on lamplighter groups with acylindrically hyperbolic bases, including cases with infinite groups.

Findings

Proves CLT for the drift of non-elementary random walks on lamplighter groups.

Provides upper bounds on the central moments of the drift.

Extends results to arbitrary finitely generated groups A.

Abstract

We prove a Central Limit Theorem for the drift of a non-elementary random walk with a finite exponential moment on a wreath product $A\wr H=\bigoplus_{H} A\rtimes H$ with $A$ a non-trivial finite group and $H$ a finitely generated acylindrically hyperbolic group. We also provide the upper bounds on the central moments of the drift. Furthermore, our results extend to the case where $A$ is an arbitrary (possibly infinite) finitely generated group.