Topological transitivity and wandering intervals for group actions on the line $\mathbb R$

TL;DR

This paper classifies all groups based on whether they can act transitively or have wandering intervals on the real line, revealing structural properties and linking orderability with group action types.

Contribution

It proves a dichotomy for group actions on the real line, classifies several groups into these types, and connects wandering type groups with indicability.

Findings

Groups are either transitive or wandering type.

Finitely generated orderable wandering groups are indicable.

Orderable higher rank lattices are of transitive type.

Abstract

For every group $G$, we show that either $G$ has a topologically transitive action on the line $\mathbb R$ by orientation-preserving homeomorphisms, or every orientation-preserving action of $G$ on $\mathbb R$ has a wandering interval. According to this result, all groups are divided into two types: transitive type and wandering type, and the types of several groups are determined. We also show that every finitely generated orderable group of wandering type is indicable. As a corollary, we show that if a higher rank lattice $\Gamma$ is orderable, then $\Gamma$ is of transitive type.