# Measure rigidity for horospherical subgroups of groups acting on trees

## TL;DR

This paper extends classical homogeneous dynamics results to actions on trees, classifying invariant measures, proving unique ergodicity, and establishing equidistribution for horospherical subgroups in a non-linear setting.

## Contribution

It provides the first classification of invariant measures and proves unique ergodicity for horospherical actions on groups acting on trees, a non-linear analogue of classical results.

## Key findings

- Classification of invariant measures for horospherical subgroups
- Proof of unique ergodicity for cocompact lattices
- Establishment of equidistribution of large orbits

## Abstract

We investigate analogues of some of the classical results in homogeneous dynamics in non-linear setting. Let $G$ be a closed subgroup of the group of automorphisms of a biregular tree and $\Gamma<G$ a discrete subgroup. For a large class of groups $G$ we give a classification of probability measures on $G/\Gamma$ invariant under horospherical subgroups. When $\Gamma$ is a cocompact lattice, we prove unique ergodicity of the horospherical action. We prove Hedlund's theorem for geometrically finite quotients. Finally, we study equidistribution of large compact orbits.