On the structure of the commutator subgroup of certain homeomorphism groups

TL;DR

This paper investigates the algebraic structure of the commutator subgroup of certain homeomorphism groups, establishing conditions for perfectness and uniform perfectness, especially in the context of factorizable and non-fixing groups on paracompact spaces.

Contribution

It extends Ling's theorem by analyzing uniform perfectness and simplicity of commutator subgroups for specific classes of homeomorphism groups, including those on open manifolds.

Findings

$[G,G]$ is perfect for factorizable non-fixing groups

$[G,G]$ is uniformly perfect for bounded factorizable non-fixing groups

Results are illustrated with examples of homeomorphism groups

Abstract

An important theorem of Ling states that if $G$ is any factorizable non-fixing group of homeomorphisms of a paracompact space then its commutator subgroup $[G,G]$ is perfect. This paper is devoted to further studies on the algebraic structure (e.g. uniform perfectness, uniform simplicity) of $[G,G]$ and $[\tilde G,\tilde G]$, where $\tilde G$ is the universal covering group of $G$. In particular, we prove that if $G$ is bounded factorizable non-fixing group of homeomorphisms then $[G,G]$ is uniformly perfect (Corollary 3.4). The case of open manifolds is also investigated. Examples of homeomorphism groups illustrating the results are given.