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

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.