$C^1$ actions on manifolds by lattices in Lie groups

TL;DR

This paper investigates smooth actions of lattice subgroups in higher-rank Lie groups on compact manifolds, proving that under certain conditions such actions are essentially trivial or factor through finite groups.

Contribution

It extends Zimmer's conjecture to $C^1$ actions, establishing new dimensional bounds for when such actions are trivial or finite.

Findings

Actions factor through finite groups when lattice rank exceeds manifold dimension

Dimensional bounds are sharp for lattices in SL(n, R)

Provides new results for $C^1$ actions in higher-rank Lie group settings

Abstract

In this paper we study Zimmer's conjecture for $C^1$ actions of lattice subgroup of a higher-rank simple Lie group with finite center on compact manifolds. We show that when the rank of an uniform lattice is larger than the dimension of the manifold, then the action factors through a finite group. For lattices in $SL(n, \R)$, the dimensional bound is sharp.