The group of almost-periodic homeomorphisms of the real line

TL;DR

This paper investigates the structure of almost-periodic homeomorphisms of the real line, showing how group actions can be conjugated to almost-periodic actions and applying this to prove a theorem about amenable left orderable groups.

Contribution

It establishes a conjugation result for group actions on the real line and provides an alternative proof of Witte's theorem linking amenability and local indicability.

Findings

Finitely generated group actions without fixed points can be conjugated to almost-periodic actions.

Actions can be compactified to 1-dimensional laminations without fixed points.

Provides an alternative proof of Witte's theorem on amenable left orderable groups.

Abstract

We study the group of almost-periodic homeomorphisms of the real line. Our main result states that an action of a finitely generated group on the real line without global fixed point is conjugated to an almost-periodic action without almost fixed point. This is equivalent to saying that the action on the real line can be compactified to an action on a 1-dimensional lamination of a compact space, without global fixed point. As an application we give an alternative proof of Witte's theorem: an amenable left orderable group is locally indicable.