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

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.