Conjugacy in a family of free-by-cyclic groups

TL;DR

This paper investigates the conjugacy problem in a specific family of free-by-cyclic groups, demonstrating that the conjugator length is linear and providing efficient algorithms for solving conjugacy problems.

Contribution

It establishes the linearity of conjugator length and offers polynomial-time algorithms for conjugacy problems in free-by-cyclic groups with positive, polynomially growing automorphisms.

Findings

Conjugator length function is linear in these groups.

Polynomial-time solutions for conjugacy and conjugacy search problems.

Analysis of the geometry and complexity of the conjugacy problem.

Abstract

We analyse the geometry and complexity of the conjugacy problem in a family of free-by-cyclic groups $H_m=F_m\rtimes\mathbb{Z}$ where the defining free-group automorphism is positive and polynomially growing. We prove that the conjugator length function of $H_m$ is linear, and describe polynomial-time solutions to the conjugacy problem and conjugacy search problem in $H_m$.