Actions of finitely generated groups on compact metric spaces

TL;DR

This paper proves that any action of a finitely generated group on a compact metric space can be realized with a compatible metric making the action bi-Lipschitz, enhancing understanding of group actions in metric geometry.

Contribution

It establishes that any group action by homeomorphisms can be metrically refined to a bi-Lipschitz action, providing a new perspective on the structure of such actions.

Findings

Existence of a compatible metric making the action bi-Lipschitz

Extension of group actions to metric settings with stronger regularity

Bridging topological and metric properties of group actions

Abstract

Let $\Gamma$ be a finitely generated group which admits an action by homeomorphisms on a compact metrizable space $X$. We show that there is a metric on $X$ defining the original topology such that for this metric, the action is by bi-Lipschitz transformations.