Quasimorphisms on nonorientable surface diffeomorphism groups

TL;DR

This paper extends the construction of infinitely many quasimorphisms from orientable to nonorientable surface diffeomorphism groups, showing their infinite-dimensional space and implications for group properties.

Contribution

It proves the existence of infinitely many nontrivial quasimorphisms on the identity component of diffeomorphism groups of nonorientable surfaces, generalizing prior results.

Findings

The space of nontrivial quasimorphisms is infinite-dimensional.

Unboundedness of the commutator length on the diffeomorphism group.

Unboundedness of the fragmentation length on the diffeomorphism group.

Abstract

Bowden, Hensel, and Webb constructed infinitely many quasimorphisms on the diffeomorphism groups of orientable surfaces. In this paper, we extend their result to nonorientable surfaces. Namely, we prove that the space of nontrivial quasimorphisms $\widetilde{QH}(\mathrm{Diff}_0(N_g))$ on the identity component of the diffeomorphism group $\mathrm{Diff}_0(N_g)$ on a closed nonorientable surface $N_g$ of genus $g\geq 3$ is infinite-dimensional. As a corollary, we obtain the unboundedness of the commutator length and the fragmentation length on $\mathrm{Diff}_0(N_g)$.