# Conformal actions of higher rank lattices on compact pseudo-Riemannian manifolds

## TL;DR

This paper studies how higher-rank lattice groups can act conformally on compact pseudo-Riemannian manifolds, establishing bounds on lattice rank and conditions for conformal flatness, inspired by Zimmer's conjecture breakthroughs.

## Contribution

It provides new bounds on the real-rank of lattices acting conformally and proves conformal flatness when the rank is maximal, extending previous results to exceptional groups.

## Key findings

- Bound on the real-rank of lattices for conformal actions
- Manifolds are conformally flat when the lattice rank is maximal
- Improved estimates for actions of exceptional groups

## Abstract

We investigate conformal actions of cocompact lattices in higher-rank simple Lie groups on compact pseudo-Riemannian manifolds. Our main result gives a general bound on the real-rank of the lattice, which was already known for the action of the full Lie group by a result of Zimmer. When the real-rank is maximal, we prove that the manifold is conformally flat. This indicates that a global conclusion similar to that of anterior works of Bader, Nevo and Frances, Zeghib in the case of a Lie group action might be obtained. We also give better estimates for actions of cocompact lattices in exceptional groups. Our work is strongly inspired by the recent breakthrough of Brown, Fisher and Hurtado on Zimmer's conjecture.