Time complexity of the conjugacy problem in relatively hyperbolic groups

TL;DR

This paper extends known polynomial-time solutions for the conjugacy problem from hyperbolic groups to relatively hyperbolic groups, showing that conjugacy problems remain efficiently solvable under broader conditions.

Contribution

It proves that the conjugacy and conjugacy search problems are solvable in polynomial time in relatively hyperbolic groups, assuming solvability in their parabolic subgroups, and establishes linear bounds for conjugating element lengths.

Findings

Conjugacy problem solvable in polynomial time in relatively hyperbolic groups.

Linear bounds on conjugating element lengths for hyperbolic elements.

Polynomial-time solutions depend on parabolic subgroup solvability.

Abstract

If $u$ and $v$ are two conjugate elements of a hyperbolic group then the length of a shortest conjugating element for $u$ and $v$ can be bounded by a linear function of the sum of their lengths, as was proved by Lysenok. Bridson and Haefliger showed that in a hyperbolic group the conjugacy problem can be solved in polynomial time. We extend these results to relatively hyperbolic groups. In particular, we show that both the conjugacy problem and the conjugacy search problem can be solved in polynomial time in a relatively hyperbolic group, whenever the corresponding problem can be solved in polynomial time in each parabolic subgroup. We also prove that if $u$ and $v$ are two conjugate hyperbolic elements of a relatively hyperbolic group then the length of a shortest conjugating element for $u$ and $v$ is linear in terms of their lengths.