Group actions on dendrites and curves

TL;DR

This paper investigates how groups can act on dendrites and curves, providing obstructions and structural results that limit possible actions, especially for lattices in higher rank Lie groups.

Contribution

It establishes new obstructions for group actions on dendrites and extends some results to more general topological curves.

Findings

Lattices in higher rank simple Lie groups always fix a point or a pair when acting on dendrites.

A Tits alternative is proved for these group actions.

The paper describes the topological dynamics of such actions.

Abstract

We establish obstructions for groups to act by homeomorphisms on dendrites. For instance, lattices in higher rank simple Lie groups will always fix a point or a pair. The same holds for irreducible lattices in products of connected groups. Further results include a Tits alternative and a description of the topological dynamics. We briefly discuss to what extent our results hold for more general topological curves.