A Polynomial Time Algorithm For The Conjugacy Decision and Search Problems in Free Abelian-by-Infinite Cyclic Groups

TL;DR

This paper presents a polynomial time algorithm for solving conjugacy decision and search problems in free abelian-by-infinite cyclic groups, extending previous work and utilizing orbit problem algorithms.

Contribution

It introduces the first polynomial time algorithms for these problems in free abelian-by-infinite cyclic groups, adapting existing methods and applying orbit problem solutions.

Findings

Polynomial time algorithm for conjugacy decision problem

Polynomial time algorithm for conjugacy search problem

Extension of previous algorithms to new group class

Abstract

In this paper we introduce a polynomial time algorithm that solves both the conjugacy decision and search problems in free abelian-by-infinite cyclic groups where the input is elements in normal form. We do this by adapting the work of Bogopolski, Martino, Maslakova, and Ventura in \cite{bogopolski2006conjugacy} and Bogopolski, Martino, and Ventura in \cite{bogopolski2010orbit}, to free abelian-by-infinite cyclic groups, and in certain cases apply a polynomial time algorithm for the orbit problem over $\Z^n$ by Kannan and Lipton.