Detecting conjugacy stability of subgroups in certain classes of groups

TL;DR

This paper develops effective procedures to determine conjugacy stability of subgroups across various classes of groups, including free groups, hyperbolic groups, and limit groups, advancing understanding of subgroup behavior.

Contribution

It introduces new algorithms for testing conjugacy stability in free groups, hyperbolic groups, and limit groups, extending previous theoretical results.

Findings

Effective procedure for free groups

Generalization to hyperbolic groups

Application to limit groups

Abstract

In this paper we consider the {\em conjugacy stability} property of subgroups and provide effective procedures to solve the problem in several classes of groups. In particular, we start with free groups, that is, we give an effective procedure to find out if a finitely generated subgroup of a free group is conjugacy stable. Then we further generalize this result to quasi-convex subgroups of torsion-free hyperbolic groups and finitely generated subgroups of limit groups.