Cyclic rewriting and conjugacy problems

TL;DR

This paper develops algorithms for solving conjugacy problems in groups using cyclic rewriting techniques, with applications to various complex group structures like free products and HNN-extensions.

Contribution

It introduces a cyclic rewriting framework for conjugacy problems and applies it to universal groups of Stallings pregroups, simplifying existing algorithms and proofs.

Findings

Algorithms for conjugacy in free products with amalgamation

Simplified proofs of conjugacy criteria in complex groups

Effective methods for minimal length conjugacy class representatives

Abstract

Cyclic words are equivalence classes of cyclic permutations of ordinary words. When a group is given by a rewriting relation, a rewriting system on cyclic words is induced, which is used to construct algorithms to find minimal length elements of conjugacy classes in the group. These techniques are applied to the universal groups of Stallings pregroups and in particular to free products with amalgamation, HNN-extensions and virtually free groups, to yield simple and intuitive algorithms and proofs of conjugacy criteria.