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 $\mathcal{T}}(A_Γ)$ of $Aut(A_{\Gamma })$. We answer a question of Day relating to the abelianisation of $\mathcal{T}}(A_Γ)$, and show that $\mathcal{T}}(A_Γ)$ and its image in $Out(A_{\Gamma })$ are residually torsion-free nilpotent.

arXiv comment21 pages, 1 figure. Final draft. To appear in the Journal of the London Mathematical Society

Cited by in corpus (3)

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• Outer actions of $\mathrm{Out}(F_n)$ on small right-angled Artin groups