Right-angled Artin groups of large girth and finite volume hyperbolic $3$--manifold groups

Abstract

Let $Γ$ be a finite simplicial graph of girth at least five. In this short note, we give a proof that if $M$ is a finite volume hyperbolic $3$--manifold, then the right-angled Artin group $A(Γ)$ cannot contain $π_1(M)$ as a subgroup; the argument is elementary, modulo the resolution of the Virtual Fibering Conjecture and a splitting theorem due to Belegradek. In particular, if $C_n$ denotes the $n$--cycle then $A(C_n)$ cannot contain a finite volume hyperbolic $3$--manifold group for any $n\geq 3$, thus answering a question of A.~Reid.