Large scale geometry of homeomorphism groups

TL;DR

This paper investigates the large-scale geometric structure of the identity component of the homeomorphism group of a compact manifold, linking it to the manifold's topology and dynamics of group actions.

Contribution

It establishes a well-defined quasi-isometry type for the homeomorphism group and explores its large-scale geometry through various examples.

Findings

The identity component of homeomorphism groups has a well-defined quasi-isometry type.

Large-scale geometry relates to the topology and dynamics of the manifold.

Provides examples of non-locally compact groups with applicable large-scale methods.

Abstract

Let M be a compact manifold. We show the identity component $\mathrm{Homeo}_0(M)$ of the group of self-homeomorphisms of M has a well-defined quasi-isometry type, and study its large scale geometry. Through examples, we relate this large scale geometry to both the topology of M and the dynamics of group actions on M. This gives a rich family of examples of non-locally compact groups to which one can apply the large-scale methods developed in previous work of the second author.