The monomorphism problem in free groups

TL;DR

This paper proves that the monomorphism problem in free groups is decidable and provides an effective algorithm to determine whether a monomorphism exists between two elements.

Contribution

The paper establishes the decidability of the monomorphism problem in free groups and offers a constructive algorithm for solving it.

Findings

The monomorphism problem in free groups is decidable.

An effective algorithm for the monomorphism problem is provided.

The problem's decidability was previously unknown.

Abstract

Let $F$ be a free group of finite rank. We say that the monomorphism problem in $F$ is decidable if for any two elements $u$ and $v$ in $F$, there is an algorithm that determines whether there exists a monomorphism of $F$ that sends $u$ to $v$. In this paper we show that the monomorphism problem is decidable and we provide an effective algorithm that solves the problem.