| 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. |
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