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arXiv · 1006.3269

On the structure of the commutator subgroup of certain homeomorphism groups

Ilona Michalik·Tomasz Rybicki
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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.

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Ilona Michalik, Tomasz Rybicki. 2010-06-16. On the structure of the commutator subgroup of certain homeomorphism groups. https://arxiv.org/abs/1006.3269

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