Almost-periodic action on the real line

Abstract

A homeomorphism of the real line is almost-periodic if the set of its conjugates by the translations is relatively compact in the compact open topology. Our main result states that an action of a finitely generated group on the real line without global fixed points is conjugated to an action by almost-periodic homeomorphisms without almost fixed points. This is equivalent to saying that the real line together with the translation flow can be compactified as an orbit of a free action of $\mathbb{R}$ on a compact space, together with an action of the group by homeomorphisms withou global fixed points. As an application we give an alternative proof of Witte’s theorem: an amenable left orderable group is locally indicable.