# On factor rigidity and joining classification for infinite volume rank one homogeneous spaces

## TL;DR

This paper classifies joinings with respect to the Burger-Roblin measure for horospherical actions on certain infinite volume hyperbolic manifolds, revealing rigidity phenomena in these dynamical systems.

## Contribution

It provides a classification of locally finite joinings and establishes rigidity of certain set-valued maps for infinite volume rank one homogeneous spaces.

## Key findings

- Classification of joinings with respect to Burger-Roblin measure
- Rigidity of U-equivariant set-valued maps in geometrically finite cases
- Extension of rigidity results to broader classes of infinite volume hyperbolic manifolds

## Abstract

We classify locally finite joinings with respect to the Burger-Roblin measure for the action of a horospherical subgroup $U$ on $\Gamma \backslash G$, where $G = \operatorname{SO}(n,1)^\circ$ and $\Gamma$ is a convex cocompact and Zariski dense subgroup of $G$, or geometrically finite with restrictions on critical exponent and rank of cusps. We also prove in the more general case of $\Gamma$ geometrically finite and Zariski dense that certain $U$-equivariant set-valued maps are rigid.