Conjugacy in normal subgroups of hyperbolic groups

TL;DR

This paper investigates the conjugacy problem in normal subgroups of hyperbolic groups, establishing criteria for solvability and separability, and provides new examples of conjugacy separable groups with non-separable subgroups.

Contribution

It introduces criteria linking conjugacy properties of normal subgroups to their quotients and constructs novel examples of conjugacy separable groups with non-separable subgroups.

Findings

Normal subgroups inherit conjugacy properties from quotients under certain conditions

Hyperbolic groups from Rips's construction are hereditarily conjugacy separable

Existence of finitely generated, finitely presented conjugacy separable groups with non-conjugacy separable subgroups

Abstract

Let N be a finitely generated normal subgroup of a Gromov hyperbolic group G. We establish criteria for N to have solvable conjugacy problem and be conjugacy separable in terms of the corresponding properties of G/N. We show that the hyperbolic group from F. Haglund's and D. Wise's version of Rips's construction is hereditarily conjugacy separable. We then use this construction to produce first examples of finitely generated and finitely presented conjugacy separable groups that contain non-(conjugacy separable) subgroups of finite index.