Commutators of contactomorphisms

TL;DR

This paper proves that the identity component of the contactomorphism group of a connected manifold is perfect and simple, resolving a long-standing problem in the algebraic structure of contactomorphism groups.

Contribution

It establishes the perfectness and simplicity of the contactomorphism group, a key algebraic property previously unknown for this classical group.

Findings

The contactomorphism group is perfect.

The contactomorphism group is simple.

Addresses a long-standing open problem.

Abstract

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.