Quantifying separability in virtually special groups

Mark F. Hagen and Priyam Patel

Vol. 284 (2016), No. 1, 103–120

DOI: 10.2140/pjm.2016.284.103

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 $Q\le G$ 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 (Math. Z. 279 (2015), 297–310) and a result of Patel on surface groups (Proc. Amer. Math. Soc. 142 (2014), 2891–2906).

Keywords

subgroup separable, right-angled Artin groups, quantifying, virtually special groups

Mathematical Subject Classification 2010

Primary: 20E26

Secondary: 20F36

Milestones

Received: 7 August 2015

Revised: 29 September 2015

Accepted: 3 February 2016

Published: 10 July 2016