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

TL;DR

This paper explores the constraints on higher-rank lattices acting on right-angled Artin groups, using Johnson homomorphisms and the structure of automorphism groups to establish bounds and properties.

Contribution

It provides an upper bound on the real rank of Lie groups based on clique structures in the associated Artin groups and answers a key question about the Torelli subgroup's abelianisation.

Findings

Bound on the real rank of G determined by clique structure in Γ

Torelli subgroup and its image are residually torsion-free nilpotent

Answer to Day's question on the abelianisation of the Torelli subgroup

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.