On subgroup conjugacy separability of hyperbolic QVH-groups

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, H1 is a quasiconvex subgroup of G, and H2 is a subgroup of G which is elementwise conjugate into H1, then there exists a finite index subgroup of H2 which is conjugate into H1. 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.This property logically continues the following series of well known properties of groups: residual finiteness, conjugacy separability (CS), and subgroup separability (LERF). Note that SCS-groups are residually finite, but there are residually finite, and even conjugacy separable groups, which are not SCS-groups. The SCS-property is relatively new and not much is known about, which groups enjoy this property.In [16], Grunewald and Segal proved that all virtually polycyclic groups are SCS (see also Theorem 7 in Chapter 4 of [39]). In the preprint [6], Bogopolski and Grunewald proved that free groups and some virtually free groups are SCS. In the preprint [2], Bogopolski and Bux proved that the fundamental groups of closed orientable surfaces are SCS. Chagas and Zalesskii [9] generalized this to limit groups. To formulate our results, we need the following definitions:

Hyperbolic 3-Manifolds Groups are Subgroup Conjugacy Separable

“…The objective of this paper is to show that the fundamental group of a hyperbolic 3-manifold is subgroup conjugacy separable. Bogopolski and Bux posed it as an open question on page 3 in [6].…”