On the structure of the commutator subgroup of certain homeomorphism groups

Details On the structure of the commutator subgroup of certain homeomorphism groups On the structure of the commutator subgroup of certain homeomorphism groups

2010-06-16

arxiv.org/abs/1006.3269v1

Topol. Appl. 158(2011), 1314-1324

Mathematics

Differential Geometry

18 pages

Scientific paper

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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