Isoperimetric profile and random walks on locally compact solvable groups

TL;DR

This paper investigates the isoperimetric profiles of a broad class of amenable locally compact solvable groups, revealing their optimality and implications for random walk behavior and geometric properties.

Contribution

It characterizes the isoperimetric profile of these groups and connects it to random walk return probabilities and unique geometric features.

Findings

Isoperimetric profile is optimal among amenable groups.

Computed return probabilities for symmetric random walks.

Identified geometric properties specific to these groups.

Abstract

We study a large class of amenable locally compact groups containing all solvable algebraic groups over a local field and their discrete subgroups. We show that the isoperimetric profile of these groups is in some sense optimal among amenable groups. We use this fact to compute the probability of return of symmetric random walks, and to derive various other geometric properties which are likely to be only satisfied by these groups.