Random walks on the discrete affine group

TL;DR

This paper studies random walks on the discrete affine group of a regular tree, describing its boundary, isoperimetric profile, and metric properties, and relating it to lamplighter groups.

Contribution

It introduces the discrete affine group of a regular tree, characterizes its Poisson boundary, and computes key geometric and probabilistic invariants, linking it to lamplighter groups.

Findings

Poisson boundary described as a configuration space

Computed isoperimetric profile and Hilbert compression exponent

Established metric relationships with lamplighter groups

Abstract

We introduce the discrete affine group of a regular tree as a finitely generated subgroup of the affine group. We describe the Poisson boundary of random walks on it as a space of configurations. We compute isoperimetric profile and Hilbert compression exponent of the group. We also discuss metric relationship with some lamplighter groups and lamplighter graphs.