On the structure of the commutator subgroup of certain homeomorphism groups

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 [ G ˜ , G ˜ ] , where G ˜ is the universal covering group of G. In particular, we prove that if G is a 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.