Free-by-cyclic groups have solvable conjugacy problem

TL;DR

This paper proves that the conjugacy problem is solvable in free-by-cyclic groups by developing algorithms that utilize automorphism fixed subgroups and power mappings, enabling effective conjugacy and Reidemeister equivalence detection.

Contribution

It introduces an algorithmic solution for the conjugacy problem in free-by-cyclic groups, building on prior automorphism fixed subgroup and power mapping results.

Findings

Conjugacy problem is solvable in free-by-cyclic groups.

Algorithms can determine conjugacy and Reidemeister equivalence.

Effective computation of conjugating elements is achieved.

Abstract

We show that the conjugacy problem is solvable in [finitely generated free]-by-cyclic groups, by using a result of O. Maslakova that one can algorithmically find generating sets for the fixed subgroups of free group automorphisms, and one of P. Brinkmann that one can determine whether two cyclic words in a free group are mapped to each other by some power of a given automorphism. The algorithm effectively computes a conjugating element, if it exists. We also solve the power conjugacy problem and give an algorithm to recognize if two given elements of a finitely generated free group are Reidemeister equivalent with respect to a given automorphism.