Rigidity for higher rank lattice actions on dendrites

TL;DR

This paper investigates the rigidity of higher rank lattice actions on dendrites, showing restrictions on their dynamics and establishing new characterizations of group orderability through these actions.

Contribution

It provides new rigidity results for higher rank lattice actions on dendrites and characterizes left-orderability of groups via dendrite actions.

Findings

Higher rank lattice actions on dendrites cannot be almost free.

Actions of certain lattices have nontrivial almost finite subsystems.

A new characterization of left-orderability via dendrite actions.

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 $SL_n(\mathbb Z)$ with $n\geq 3$, 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.