Quantifying separability in virtually special groups

TL;DR

This paper provides an effective proof of subgroup separability in special groups, demonstrating polynomial and exponential bounds on the index of separating subgroups for quasiconvex elements.

Contribution

It introduces a new proof technique for subgroup separability in virtually special hyperbolic groups, extending previous residual finiteness results.

Findings

Separable subgroups have polynomial index bounds for quasiconvex elements.

Generalizes residual finiteness growth results to broader classes of groups.

Provides effective methods for subgroup separation in special groups.

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.