Rigidity for non-left-orderable group actions on dendrites

Abstract

We establish two rigidity theorems for non-left-orderable group actions on dendrites. Explicitly, letting $X$ be a dendrite without infinite order points, we show that if $\Gamma$ is a non-virtually left-orderable small group, then no $\Gamma$-action on $X$ is almost free, and if $\Gamma$ is a higher-rank lattice, then every $\Gamma$-action on $X$ is a highly proximal extension of an almost finite action.