Actions of higher-rank lattices on free groups

TL;DR

This paper proves that higher-rank lattices in semisimple Lie groups have only finite actions on the outer automorphism groups of finitely generated free groups, revealing rigidity in their representations.

Contribution

It establishes a new rigidity result showing that such lattices cannot have infinite images in the outer automorphism group of free groups.

Findings

Homomorphisms from higher-rank lattices to Out(F) have finite image

Higher-rank lattices exhibit rigidity in their actions on free groups

No infinite representations of these lattices into Out(F)

Abstract

If $G$ is a semisimple Lie group of real rank at least 2 and $\Gamma$ is an irreducible lattice in $G$, then every homomorphism from $\Gamma$ to the outer automorphism group of a finitely generated free group has finite image.