On subgroup conjugacy separability of hyperbolic QVH-groups

TL;DR

This paper proves subgroup conjugacy separability properties for hyperbolic groups, including fundamental groups of hyperbolic 3-manifolds and small cancellation groups, establishing new separability results for these classes.

Contribution

It establishes that hyperbolic groups with quasiconvex subgroups are hereditarily quasiconvex-SCS and SICS, extending subgroup conjugacy separability to several important classes of groups.

Findings

Hyperbolic groups with quasiconvex subgroups are hereditarily quasiconvex-SCS and SICS.

Surface groups are shown to be SCS and SICS.

Certain small cancellation groups cannot have the quasiconvex condition removed.

Abstract

A group $G$ is called subgroup conjugacy separable (abbreviated as SCS) if any two finitely generated and non-conjugate subgroups of $G$ remain non-conjugate in some finite quotient of $G$. An into-conjugacy version of SCS is abbreviated by SICS. We prove that if $G$ is a hyperbolic group, $H_1$ is a quasiconvex subgroup of $G$, and $H_2$ is a subgroup of $G$ which is elementwise conjugate into $H_1$, then there exists a finite index subgroup of $H_2$ which is conjugate into $H_1$. As corollary, we deduce that fundamental groups of closed hyperbolic 3-manifolds and torsion-free small cancellation groups with finite $C'(1/6)$ or $C'(1/4)-T(4)$ presentations are hereditarily quasiconvex-SCS and hereditarily quasiconvex-SICS, and that surface groups are SCS and SICS. We also show that the word "quasiconvex" cannot be deleted for at least small cancellation groups.