On the conjugacy problem for subdirect products of hyperbolic groups

TL;DR

This paper characterizes when the conjugacy problem in finitely generated subdirect products of torsion-free hyperbolic groups is solvable, linking it to membership decision algorithms for cyclic subgroups in certain quotients.

Contribution

It introduces a new technique for perturbing elements in hyperbolic groups to prevent proper powers, enabling a characterization of the conjugacy problem's solvability.

Findings

Conjugacy problem solvability is equivalent to a membership decision problem.

A new perturbation technique for hyperbolic group elements is developed.

The result applies to finitely generated subdirect products of torsion-free hyperbolic groups.

Abstract

If $G_1$ and $G_2$ are torsion-free hyperbolic groups and $P<G_1\times G_2$ is a finitely generated subdirect product, then the conjugacy problem in $P$ is solvable if and only if there is a uniform algorithm to decide membership of the cyclic subgroups in the finitely presented group $G_1/(P\cap G_1)$. The proof of this result relies on a new technique for perturbing elements in a hyperbolic group to ensure that they are not proper powers.