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

TL;DR

This paper explores the conjugacy problem in fibre products of torsion-free hyperbolic groups, linking conjugator length to subgroup geometry, Dehn functions, and distortion, thus expanding understanding of conjugator length functions.

Contribution

It establishes inequalities connecting conjugator length with geometric and algebraic properties of fibre products, providing new tools for analyzing conjugacy complexity.

Findings

Derived bounds relating conjugator length to cyclic subgroup geometry

Connected conjugator length to Dehn functions and distortion measures

Extended the class of functions known as conjugator length functions

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 {\em 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.