Computing JSJ decompositions of hyperbolic groups

TL;DR

This paper introduces an algorithm to compute the JSJ decomposition of one-ended hyperbolic groups over virtually cyclic subgroups, using boundary topology and Cayley graph geometry, avoiding Makanin's algorithm.

Contribution

It provides a new geometric approach to compute JSJ decompositions and determine virtual Fuchsian groups without relying on Makanin's algorithm.

Findings

Algorithm computes JSJ decompositions efficiently.

New method identifies virtual Fuchsian groups.

Avoids complex existing algorithms like Makanin's.

Abstract

We present an algorithm that computes Bowditch's canonical JSJ decomposition of a given one-ended hyperbolic group over its virtually cyclic subgroups. The algorithm works by identifying topological features in the boundary of the group. As a corollary we also show how to compute the JSJ decomposition of such a group over its virtually cyclic subgroups with infinite centre. We also give a new algorithm that determines whether or not a given one-ended hyperbolic group is virtually fuchsian. Our approach uses only the geometry of large balls in the Cayley graph and avoids Makanin's algorithm.