Commutator length of annulus diffeomorphisms

TL;DR

This paper investigates the structure of the diffeomorphism group of the annulus, showing it has a one-dimensional space of homogeneous quasi-morphisms and an unbounded commutator length, contributing to understanding of diffeomorphism group properties.

Contribution

It demonstrates that the group of annulus diffeomorphisms has a one-dimensional space of homogeneous quasi-morphisms and unbounded commutator length, providing a new example in the study of diffeomorphism groups.

Findings

The space of homogeneous quasi-morphisms is one-dimensional for r ≠ 3.

The commutator length on this group is unbounded.

Provides an example of a manifold with an unbounded diffeomorphism group.

Abstract

We study the group of C^{r}-diffeomorphisms of the closed annulus that are isotopic to the identity. We show that, for r different from 3, the linear space of homogeneous quasi-morphisms on this group is one dimensional. Therefore, the commutator length on this group is (stably) unbounded. In particular, this provides an example of a manifold whose diffeomorphisms group is unbounded in the sense of Burago, Ivanov and Polterovich.