Quantifying separability in RAAGs via representations

Abstract

We answer the question raised by Louder, McReynolds and Patel in their 2017 article and prove the following statement. Let $L$ be a right-angled Artin group (abbreviated as RAAG) and $H$ a word quasiconvex subgroup of $L$; then there is a finite-dimensional representation of $L$ that separates the subgroup $H$ in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate $H$ in $L$. This implies the same statement for a virtually special group $L$, and, in particular, for the fundamental groups of a hyperbolic $3$-manifold.