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

Let $\mathrm{\Gamma }$ 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 $A(\mathrm{\Gamma })$ cannot contain ${\pi }_1(M)$ 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 $n\ge 3$, thus answering a question of A.~Reid.