Arithmetic quotients of the automorphism group of a right-angled Artin group

Abstract

It was previously shown by Grunewald and Lubotzky that the automorphism group of a free group, $\operatorname{Aut}(F\_n)$, has a large collection of virtual arithmetic quotients. Analogous results were proved for the mapping class group by Looijenga and by Grunewald, Larsen, Lubotzky, and Malestein. In this paper, we prove analogous results for the automorphism group of a right-angled Artin group for a large collection of defining graphs. As a corollary of our methods we produce new virtual arithmetic quotients of $\operatorname{Aut}(F\_n)$ for $n \geq 4$ where $k$th powers of all transvections act trivially for some fixed $k$. Thus, for some values of $k$, we deduce that the quotient of $\operatorname{Aut}(F\_n)$ by the subgroup generated by $k$th powers of transvections contains nonabelian free groups. This expands on results of Malestein and Putman and of Bridson and Vogtmann.