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.