On distortion in groups of homeomorphisms

TL;DR

This paper studies the distortion properties of homeomorphisms on topological spaces, introducing invariants like invariant measures and rotation numbers to determine when such homeomorphisms are undistorted.

Contribution

It introduces a Nielsen-type invariant measure relation and a local rotation number to characterize undistorted elements in the homeomorphism group.

Findings

Homeomorphisms with two nonequivalent invariant measures are undistorted.

A nonconstant rotation number implies the homeomorphism is undistorted.

Abstract

Let X be a path-connected topological space admitting a universal cover. Let Homeo(X,a) denote the group of homeomorphisms of X preserving degree one cohomology class a. We investigate the distortion in Homeo(X,a). Let g be an element of Homeo(X,a). We define a Nielsen-type equivalence relation on the space of g-invariant Borel probability measures on X and prove that if a homeomorphism g admits two nonequivalent invariant measures then it is undistorted. We also define a local rotation number of a homeomorphism generalising the notion of the rotation of a homeomorphism of the circle. Then we prove that a homeomorphism is undistorted if its rotation number is nonconstant.