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 resp. in 1970s. The results and the methods of their proofs are similar to each other.

In this talk I present a solution of the long-standing problem on the algebraic structure of the contactomorphism group. Namely, I show that the compactly supported identity component of it is perfect and simple (if the underlying manifold is connected). In the proof, completely different than those in the previous cases, I use well-known facts (a fixed point theorem, the Lychagin chart) as well as some new ideas (fragmentations of the second kind, new rolling-up operators).

Possible applications of the result are indicated. E.g. the result is connected with the notion of the commutator length for the contactomorphism group. In turn, in analogy to the symplectic case, the commutator length may be related to some deep facts and problems in the contact geometry.