The conjugacy problem in hyperbolic groups for finite lists of group elements

TL;DR

This paper presents a theoretical algorithm for solving the conjugacy problem in hyperbolic groups, capable of determining conjugacy of finite lists of elements and computing centralisers efficiently, given pre-computed structures.

Contribution

It introduces a non-practical but theoretically efficient algorithm for conjugacy and centraliser problems in hyperbolic groups with explicit time bounds.

Findings

Algorithm determines conjugacy of lists in O(m mu) time

Computes generators of centralisers within the same time bound

Provides a theoretical framework for conjugacy problems in hyperbolic groups

Abstract

Let G be a word-hyperbolic group with given finite generating set, for which various standard structures and constants have been pre-computed. A (non-practical) algorithm is described that, given as input two lists A and B, each composed of m words in the generators and their inverses, determines whether or not the lists are conjugate in G, and returns a conjugating element should one exist. The algorithm runs in time O(m mu)$, where mu is an upper bound on the lengths of elements in the two lists. Similarly, an algorithm is outlined that computes generators of the centraliser of A, with the same bound on running time.