Test elements in torsion-free hyperbolic groups

TL;DR

This paper characterizes test elements in torsion-free hyperbolic groups, showing they are precisely those not contained in any proper retract, thus providing a clear criterion for identifying such elements.

Contribution

It establishes a complete characterization of test elements in torsion-free hyperbolic groups, linking them to the absence from proper retracts.

Findings

Test elements are exactly those not contained in proper retracts.

Provides a criterion for identifying test elements in torsion-free hyperbolic groups.

Enhances understanding of subgroup structure in hyperbolic groups.

Abstract

We prove that in a torsion-free hyperbolic group, an element is a test element if and only if it is not contained in a proper retract.