Smoothing countable group actions on metrizable spaces

TL;DR

This paper demonstrates that any topological action of a countable group on a metrizable space can be represented as a bi-Lipschitz action with an appropriate metric, extending previous results and providing new proofs for related theorems.

Contribution

It extends Hamenstädt's result to all countable groups and offers a simplified proof of a theorem on one-manifolds, also establishing analogous results for subgroups of locally compact groups.

Findings

Every topological countable group action can be realized as bi-Lipschitz.

Provides a simplified proof of a theorem on one-manifolds.

Establishes analogous results for subgroups of locally compact groups.

Abstract

We prove that every topological action of a countable group on a metrizable space can be realized as a bi-Lipschitz action with respect to some compatible metric. This extends a result due to U. Hamenst\"{a}dt regarding finitely generated groups, and our proof is based upon her idea. This also gives a simple proof of a theorem due to Deroin, Kleptsyn and Navas regarding one-manifolds. We also establish an analogous result for closed subgroups of locally compact groups.