The Burnside problem for locally compact groups

TL;DR

This paper proves that a locally compact group is non-compact if and only if it admits a translation-like action by the integers, extending geometric group theory concepts to topological groups.

Contribution

It establishes a topological version of the Burnside problem for locally compact groups and characterizes translation-like actions of nd nd free groups.

Findings

A locally compact group is non-compact iff it admits a ction.

Characterization of cocompact translation-like actions on locally compact groups.

Generalization of previous results by Schneider and Seward.

Abstract

Using topological notions of translation-like actions introduced by Schneider, we give a positive answer to a geometric version of Burnside problem for locally compact group. The main theorem states that a locally compact group is non-compact if and only if it admits a translation-like action by the group of integers $\mathbf Z$. We then characterize the existence of cocompact translation-like actions of $\mathbf Z$ or non-abelian free groups on a large class of locally compact groups, improving on Schneider's results and generalising Seward's.