# Quantifying separability in virtually special groups

Mark F. Hagen, Priyam Patel

Research output: Contribution to journalArticle (Academic Journal)peer-review

## Abstract

We give a new, effective proof of the separability of cubically convex-cocompact subgroups of special groups. As a consequence, we show that if $G$ is a virtually compact special hyperbolic group, and $Q\leq G$ is a $K$-quasiconvex subgroup, then any $g\in G-Q$ of word-length at most $n$ is separated from $Q$ by a subgroup whose index is polynomial in $n$ and exponential in $K$. This generalizes a result of Bou-Rabee and the authors on residual finiteness growth and a result of the second author on surface groups.
Original language English 103-120 Pacific Journal of Mathematics 284 1 https://doi.org/10.2140/pjm.2016.284.103 Published - 10 Jul 2016

### Bibliographical note

12 pages, 5 figures. Revised in light of referee's comments. To appear in Pacific J. Math

## Keywords

• math.GR
• math.GT
• 20E26, 20F36
• subgroup separable
• right-angled Artin groups
• quantifying
• virtually special groups