# Boundary actions of lattices and $C^0$ local semi-rigidity

## TL;DR

This paper studies the stability of boundary actions of lattices in semisimple Lie groups, showing that small perturbations are semi-conjugate to the original action, but also constructing non semi-conjugate perturbations.

## Contribution

It extends semi-rigidity results to boundary actions of lattices, demonstrating continuous semi-conjugacy under perturbations and providing counterexamples for higher regularity.

## Key findings

- Perturbations of standard boundary actions are semi-conjugate to the original.
- The semi-conjugacy is close to the identity in the homeomorphism group.
- Counterexamples show some perturbations are not semi-conjugate at the $C^0$ level.

## Abstract

We consider actions of cocompact lattices in semisimple Lie groups of the noncompact type on their boundaries $G/Q$, $Q$ a parabolic group, the so-called standard actions. We show that perturbations of the standard action in the homeomorphism group continuously factor onto the original standard action by a semi-conjugacy close to the identity. This generalizes works by Bowden, Mann, Manning and Weisman in the setting of negative curvature or Gromov hyperbolic groups. Finally, we also construct perturbations of the action of lattices on the geodesic boundary which are not $C^0$ semi-conjugate to the original action.