Quantifying separability in RAAGs via representations

TL;DR

The paper proves that for a right-angled Artin group $L$ and a word quasiconvex (in particular, cubically convex-cocompact) subgroup $H$, there exists a finite-dimensional faithful representation $\rho_H$ such that $\overline{\rho_H(H)} \cap \rho_H(L) = \rho_H(H)$, yielding an effective, polynomial-bounded separation of $H$ from $L$ in finite quotients. The authors introduce and exploit the canonical completion in special cube complexes to obtain a finite-index subgroup $K$ built via HNN-extensions, enabling a constructive representation-theoretic separation that extends from $K$ to $L$ by induction. They provide explicit polynomial bounds on the size of separating quotients, and show that these results extend to virtually special groups and to fundamental groups of hyperbolic 3-manifolds, with implications for effective separability in broader geometric-group-theoretic contexts. The work combines geometric, combinatorial, and algebraic techniques to translate separability into a Zariski-closure separation problem, yielding practical, quantitative control over finite quotients used for separation.

Abstract

We answer the question asked by Louder, McReynolds and Patel, and prove the following statement. Let L be a RAAG, 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, a fundamental groups of a hyperbolic 3-manifold.

Figures (1)

• Figure 1: There is a local isometry $f: Y\rightarrow S(\Gamma )$ and $C(Y)$ is the canonical completion of $Y$. Only 1-skeletons are shown.

Theorems & Definitions (17)

• Theorem 1
• Corollary 2
• Lemma 3
• proof
• Lemma 4
• Proposition 5
• Proposition 6
• proof
• Remark 7
• Remark 8