Commutators of contactomorphisms

The group of volume preserving diffeomorphisms, the group of symplectomorphisms and the group of contactomorphisms constitute the classical groups of diffeomorphisms. The first homology groups of the compactly supported identity components of the first two groups have been computed by Thurston and Banyaga, respectively. In this paper we solve the long standing problem on the algebraic structure of the third classical diffeomorphism group, i.e. the contactomorphism group. Namely we show that the compactly supported identity component of the group of contactomorphisms is perfect and simple (if the underlying manifold is connected). The result could be applied in various ways.Comment: revised version; 38 page

Isomorphisms between groups of equivariant homeomorphisms of GG-manifolds with one orbit type

Given a compact Lie group $G$, a reconstruction theorem for free $G$-manifolds is proved. As a by-product reconstruction results for locally trivial bundles are presented. Next, the main theorem is generalized to $G$-manifolds with one orbit type. These are the first reconstruction results in the category of $G$-spaces, showing also that the reconstruction in this category is very specific and involved.Comment: 18 page

On the homeomorphism groups of manifolds and their universal coverings

Let $\mathcal H_c(M)$ stand for the path connected identity component of the group of all compactly supported homeomorphisms of a manifold $M$. It is shown that $\mathcal H_c(M)$ is perfect and simple under mild assumptions on $M$. Next, conjugation-invariant norms on \H_c(M) are considered and the boundedness of $\mathcal H_c(M)$ is investigated. Finally, the structure of the universal covering group of $\mathcal H_c(M)$ is studied.Comment: 19 page

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 $[\tilde G,\tilde G]$, where $\tilde G$ 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.Comment: 18 page