Permute and conjugate: the conjugacy problem in relatively hyperbolic groups

TL;DR

This paper introduces the permutation conjugacy length function in relatively hyperbolic groups, providing bounds and applications to conjugacy problem complexity, growth rates, and language properties, extending efficient algorithms from hyperbolic groups.

Contribution

It defines and analyzes the permutation conjugacy length function in relatively hyperbolic groups, showing it is bounded by that of parabolic subgroups, with implications for conjugacy problem complexity.

Findings

Permutation conjugacy length function is bounded by that of parabolic subgroups.

Applications to conjugacy problem complexity and growth rates.

Provides bounds even when standard conjugacy length is unbounded.

Abstract

Modelled on efficient algorithms for solving the conjugacy problem in hyperbolic groups, we define and study the permutation conjugacy length function. This function estimates the length of a short conjugator between words $u$ and $v$, up to taking cyclic permutations. This function might be bounded by a constant, even in the case when the standard conjugacy length function is unbounded. We give applications to the complexity of the conjugacy problem, estimating conjugacy growth rates, and languages. Our main result states that for a relatively hyperbolic group, the permutation conjugacy length function is bounded by the permutation conjugacy length function of the parabolic subgroups.