On the structure of the commutator subgroup of certain homeomorphism groups

Abstract

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], whereG is the universal covering group of G. In particular, we prove that if G is 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. Definition 1.1. Let U be an open cover of X. A group of homeomorphisms G of a space X is called U-factorizable if for every g ∈ G there are g 1 , . . . , g r ∈ G with g = g 1 . . . g r and such that supp(g i ) ⊂ U i , i = 1, . . . , r, for some U 1 , . . . , U r ∈ U. G is called factorizable if for every open cover U of X it is U-factorizable.

Algebraic Structure of Diffeomorphism Groups of One-Manifolds

“…The remainder of this subsection is devoted to proving Theorem 3.11. Our proof is based on the ideas of Epstein, Ling and Michalik-Rybicki [38].…”

On the boundedness of equivariant homeomorphism groups

“…Next Proposition 2.6 or 2.9 applied to H(N ) implies that H(N ) is perfect. The simplicity then follows from [15] (see also [8]). …”

“…To show the second assertion we employ a reasoning from Mather [14] concerning the perfectness for H(U ) ∼ instead of H(U ), U being a ball or half-ball (see also the proof of Lemma 3.5(1) below) with some modifications as in [15]. Next it suffices to apply Corollary 2.7 or 2.10.…”

On the uniform perfectness of groups of bundle homeomorphisms

“…The problem of the algebraic structure of homeomorphism groups, especially the boundedness and uniform perfectness of them, has drawn much attention. It has been studied among others in [1], [4], [5], [6], [7], [8], [9], [11], [12], [14], [15], [16] (see also references therein).…”