Conjugacy in fibre products, distortion, and the geometry of cyclic subgroups

Martin R. Bridson

Vol. 345 (2026), No. 1, 19–44

DOI: 10.2140/pjm.2026.345.19

Abstract

We investigate the complexity of the conjugacy problem for fibre products in torsion-free hyperbolic groups. Let $G$ be a torsion-free hyperbolic group and let $P<G\times G$ be the fibre product associated to an epimorphism $G\twoheadrightarrow Q$. We establish inequalities that relate the conjugator length function of $P$ to the geometry of cyclic subgroups in $Q$, the Dehn function of $Q$, the rel-cyclics Dehn function of $Q$, and the distortion of $P$ in $G\times G$. These estimates provide tools for extending the library of (large) functions that are known to arise as the conjugator length functions of finitely generated and finitely presented groups.

Keywords

fibre products, conjugator length, cyclic subgroups, rel-cyclics Dehn function

Mathematical Subject Classification

Primary: 20F10, 20F65, 20F67

Milestones

Received: 20 July 2025

Revised: 9 July 2026

Accepted: 9 July 2026

Published: 25 September 2026