Quantifying separability in virtually 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 ≤ G is a K-quasiconvex subgroup, then any g ∈ 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 fi niteness growth and a result of the second author on surface groups.