Sebastian Hensel: Quasimorphisms on Diffeomorphism Groups (and Fine Curve Graphs)

Abstract:

It is a classical result of Thurston and Mather that the identity components of diffeomorphism groups of compact manifolds are perfect. In fact, work by Burago-Ivanov-Polterovich and Tsuboi showed that for spheres of any dimension, or arbitrary manifolds in in dimensions different from 2 or 4 they are uniformly perfect.

However, the case of surfaces of positive genus shows radically different behaviour — I will discuss joint work with Jonathan Bowden and Richard Webb which constructs unbounded quasi-morphisms on the identity component of diffeomorphism groups of surfaces, showing that they are not uniformly perfect. The tools also open an avenue to use tools from geometric group theory to study these diffeomorphism groups. Time permitting, I will indicate some of these newer results.