One Variable Equations in Torsion-Free Hyperbolic Groups

TL;DR

This paper characterizes when solutions to one-variable equations in torsion-free hyperbolic groups form a finite union of points and cosets of centralizers, linking it to the freeness of two-generator subgroups.

Contribution

It provides a complete characterization of the solution sets for one-variable equations in torsion-free hyperbolic groups based on subgroup properties.

Findings

Solution sets are finite unions of points and cosets of centralizers under certain conditions.

A key condition is that all two-generator subgroups of the group are free.

The paper establishes an equivalence between this solution structure and subgroup freeness.

Abstract

Let $\Gamma$ be a torsion-free hyperbolic group. We show that the set of solutions of any system of equations with one variable in $\Gamma$ is a finite union of points and cosets of centralizers if and only if any two-generator subgroup of $\Gamma$ is free.