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

摘要

Let G be a real semisimple Lie group with no compact factors and finite centre, and let Λ be an irreducible lattice in G. Suppose that there exists a homomorphism from Λ to the outer automorphism group of a right-angled Artin group A_Γ with infinite image. We give a strict upper bound to the real rank of G that is determined by the structure of cliques in Γ. An essential tool is the Andreadakis-Johnson filtration of the Torelli subgroup Tscr;(A_Γ) of Aut(A_Γ). We answer a question of Day relating to the abelianization of Tscr;(A _Γ), and show that Tscr;(A_Γ) and its image in Out(A_Γ) are residually torsion-free nilpotent.