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) over tilde, (G) over tilde], where (G) over tilde 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. (C) 2011 Elsevier B.V. All rights reserved.