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

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