The rate of escape of random walks on polycyclic and metabelian groups

TL;DR

This paper investigates the escape rates of random walks on polycyclic and metabelian groups, revealing invariance under generating set changes and linking escape rates to subgroup distortion and sandpile models.

Contribution

It introduces a new approach using subgroup distortion to determine escape rates and extends the concept to non-finitely generated subgroups in metabelian groups.

Findings

Escape rate is invariant under generating set changes in polycyclic groups.

A stronger subgroup distortion concept applies to non-finitely generated subgroups.

Escape rates in certain metabelian groups relate to dissipative abelian sandpile toppling.

Abstract

We use subgroup distortion to determine the rate of escape of a simple random walk on a class of polycyclic groups, and we show that the rate of escape is invariant under changes of generating set for these groups. For metabelian groups, we define a stronger form of subgroup distortion, which applies to non-finitely generated subgroups. Under this hypothesis, we compute the rate of escape for certain random walks on metabelian groups via a comparison to the toppling of a dissipative abelian sandpile.