Computing JSJ decompositions of hyperbolic groups

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.

arXiv comment38 pages, one figure. v4 corrects two spelling mistakes. This is the final version accepted by the Journal of Topology

References in corpus (5)

• JSJ decompositions of groups
• The isomorphism problem for all hyperbolic groups
• Line Patterns in Free Groups
• Detecting free splittings in relatively hyperbolic groups
• Detecting geometric splittings in finitely presented groups

Cited by in corpus (4)

• JSJ decompositions of groups
• Deciding Isomorphy using Dehn fillings, the splitting case
• Quasi-Isometries for certain Right-Angled Coxeter Groups
• Immersed cycles and the JSJ decomposition