Conjugacy classes of solutions to equations and inequations over hyperbolic groups

TL;DR

This paper investigates the structure of solutions to equations over torsion-free hyperbolic groups, introducing algorithms and concepts like immutable subgroups to classify and enumerate conjugacy classes of solutions.

Contribution

It presents a novel algorithm to determine the finiteness of conjugacy classes of solutions and introduces the concept of immutable subgroups in hyperbolic groups.

Findings

Algorithm for recognizing finitely many conjugacy classes

Introduction of immutable subgroups

Enumeration of immutable subgroups in hyperbolic groups

Abstract

We study conjugacy classes of solutions to systems of equations and inequations over torsion-free hyperbolic groups, and describe an algorithm to recognize whether or not there are finitely many conjugacy classes of solutions to such a system. The class of immutable subgroups of hyperbolic groups is introduced, which is fundamental to the study of equations in this context. We apply our results to enumerate the immutable subgroups of a torsion-free hyperbolic group.