Johnson homomorphisms and actions of higher-rank lattices on right-angled Artin groups

Abstract

Let G be a real semisimple Lie group with no compact factors and finite centre, and let $\Lambda$ be a lattice in G. Suppose that there exists a homomorphism from $\Lambda$ to the outer automorphism group of a right-angled Artin group $A_\Gamma$ with infinite image. We give an upper bound to the real rank of G that is determined by the structure of cliques in $\Gamma$. An essential tool is the Andreadakis-Johnson filtration of the Torelli subgroup $\mathcal{T}}(A_\Gamma)$ of $Aut(A_\Gamma)$. We answer a question of Day relating to the abelianisation of $\mathcal{T}}(A_\Gamma)$, and show that $\mathcal{T}}(A_\Gamma)$ and its image in $Out(A_\Gamma)$ are residually torsion-free nilpotent.