Quasi-morphisms on surface diffeomorphism groups

TL;DR

This paper demonstrates that the identity component of surface diffeomorphism groups admits numerous unbounded quasi-morphisms, revealing their non-uniform perfectness and unbounded fragmentation norm, using a hyperbolic graph construction.

Contribution

It introduces a new hyperbolic graph for surface diffeomorphism groups and shows these groups have many unbounded quasi-morphisms, answering longstanding questions.

Findings

The group is not uniformly perfect.

The fragmentation norm is unbounded.

Constructed a hyperbolic graph analogous to the curve graph.

Abstract

We show that the identity component of the group of diffeomorphisms of a closed oriented surface of positive genus admits many unbounded quasi-morphisms. As a corollary, we also deduce that this group is not uniformly perfect and its fragmentation norm is unbounded, answering a question of Burago--Ivanov--Polterovich. As a key tool we construct a hyperbolic graph on which these groups act, which is the analog of the curve graph for the mapping class group.