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

Abstract

Abstract. 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 T (A Γ ) of Aut(A Γ ). We answer a question of Day relating to the abelianisation of T (A Γ ), and show that T (A Γ ) and its image in Out(A Γ ) are residually torsion-free nilpotent.

Palindromic automorphisms of right-angled Artin groups

“…Now we recall a useful decomposition into block matrices of an image of Aut(A Γ ) inside GL(n, Z). This decomposition was observed by Day [7] and by Wade [19]. By ordering the vertices of Γ appropriately, matrices in Φ(Aut 0 (A Γ )) ≤ GL(n, Z) will have a particularly tractable lower block-triangular decomposition, which we now describe.…”

Relative automorphism groups of right‐angled Artin groups

“…One sees that the image of the abelian group in under the action on the abelianization is free abelian of rank . Furthermore, in the kernel , the fact that the remaining partial conjugations are linearly independent can be seen using the first Johnson homomorphism on (see [, Section 4]). So is isomorphic to , proving the lower bound.…”

“…Restriction maps also form an important part of the proof of the Tits alternative for (which was completed by Horbez using the work in ). Other recent results about automorphisms of RAAGs use invariance of special subgroups or restriction maps in an essential way (see, for example, ).…”

“…If the normal subgroup generated by is fixed under , then there is a projection map: which is induced by the quotient map An important feature of these maps is that each extended Laurence generator is taken either to the identity element or an extended Laurence generator of the same type under and . It is not immediate that is well defined: this is a consequence of that fact that any element of the normalizer acts by conjugation as an inner automorphism of (see [, Section 2.6] for more details).…”

Cube complexes, subgroups of mapping class groups, and nilpotent genus

“…Examples 5.1. Examples of finitely generated residually torsion-free-nilpotent groups include free groups F n (hence residually free groups such as surface groups and limit groups), right-angled Artin groups (RAAGs) [38], the Torelli subgroup of the mapping class group [5], and IA n < Out(F n ), the kernel of the natural map Out(F n ) → GL(n, Z) (see [3], [5], [30]), and the corresponding subgroup in the outer automorphism group of any RAAG [66].…”