Quasi-simplicity of some homeomorphism and diffeomorphism groups

Abstract

Abstract. The notion of quasi-simplicity of groups is introduced. It is proven that for a group of homeomorphisms G which is fixed point free and factorizable the commutator subgroup [G, G] is quasi-simple. Several examples of quasi-simple but non-simple homeomorphism groups are presented. IntroductionIn paper [8] Ling showed that for a factorizable and fixed point free group G the commutator subgroup [G, G] is the least subgroup of G, normalized by [G, G], which acts without any fixed point. In this note we introduce the notion of a quasi-simple group. This notion enables us to give a new formulation of Ling's theorem and a completely new proof of it. In the last section we present various examples of quasi-simple homeomorphism and diffeomorphism groups which are non-simple.Ling's theorem is a generalization of the following theorem of Epstein [4]: Theorem 1.1. Let X be a paracompact space, let G be a group of homeomorphisms of X and let U be a basis of open sets of X satisfying the following axioms:Axiom 3. Let g ∈ G, U ∈ U and let B ⊆ U be a covering of X. Then there exist an integer n, elements g 1 , . . . , g n ∈ G and V 1 , . . . , V n ∈ B such that g = g n g n−1 . . . g 1 , supp(g i ) ⊂ V i and supp(g i ) ∪ (g i−1 . . . g 1 (U )) = X for 1 i n.Then [G, G], the commutator subgroup of G, is simple.2000 Mathematics Subject Classification: 22A05, 22E65, 57S05.