On the structure of the commutator subgroup of certain homeomorphism groups

AbstractAn 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