\int_x^{hx}(g^*\alpha-\alpha)

TL;DR

This paper introduces a new cocycle related to group actions on topological spaces with a cohomology class, and uses it to analyze properties like distortion and rotation numbers of homeomorphisms.

Contribution

It defines a novel two-cocycle on groups acting on topological spaces and applies it to study distortion and rotation numbers of homeomorphisms.

Findings

Cocycle is cohomologically trivial if the action preserves a measure.

Homeomorphisms with non-constant local rotation number are undistorted.

Introduces a local rotation number concept for homeomorphisms.

Abstract

Let X be a connected topological space admitting a universal cover. Let a be a degree one cohomology class on X. We define and study a two-cocycle on a group acting on X by homeomorphisms preserving the class a. We use this cocycle to investigate group actions on X. For example, we show that if an action preserves a Borel probability measure on X then the cocycle is cohomologically trivial. Under various assumptions on a homeomorphism g, we prove that it is undistorted in Homeo(X,a). In particular, we introduce a local rotation number of a homeomorphism and prove that a homeomorphism with non-constant local rotation number is undistorted.