On the structure of the commutator subgroup of certain homeomorphism groups

摘要

An important theorem of Ling states that it G is any tactonzable non-hxing group of home-omorphisms of a paracompact space then its commutator subgroup [C,C] 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 homeomor-phisms 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.