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 $Q\le G$ 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 and a result of the second author on surface groups.

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

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Cited by in corpus (2)

• Residual finiteness growths of Lamplighter groups
• Explicit special covers of alternating links