Rigidity for higher rank lattice actions on dendrites

Abstract

We study the rigidity in the sense of Zimmer for higher rank lattice actions on dendrites and show that: (1) if $\Gamma$ is a higher rank lattice and $X$ is a nondegenerate dendrite with no infinite order points, then any action of $\Gamma$ on $X$ cannot be almost free; (2) if $\Gamma$ is further a finite index subgroup of $S{L}_n(\mathbb{Z})$ with $n\ge 3$, then every action of $\Gamma$ on $X$ has a nontrivial almost finite subsystem. During the proof, we get a new characterization of the left-orderability of a finitely generated group through its actions on dendrites.

arXiv commentThe second version corrected some typos and improved the expression of Theorem 1.2