On several problems about automorphisms of the free group of rank two

TL;DR

This paper presents algorithms for three automorphism-related problems in the free group of rank two, including potential positivity, bounded translation equivalence, and fixed point subgroup determination, advancing understanding of automorphism dynamics.

Contribution

It provides the first algorithms for these problems specifically in the free group of rank two, solving previously open questions.

Findings

Decidable algorithm for potential positivity of words in F2.

Algorithm to determine bounded translation equivalence in F2.

Algorithm to identify fixed point subgroups of automorphisms in F2.

Abstract

Let $F_n$ be a free group of rank $n$. In this paper we discuss three algorithmic problems related to automorphisms of $F_2$. A word $u$ of $F_n$ is called positive if $u$ does not have negative exponents. A word $u$ in $F_n$ is called potentially positive if $\phi(u)$ is positive for some automorphism $\phi$ of $F_n$. We prove that there is an algorithm to decide whether or not a given word in $F_2$ is potentially positive, which gives an affirmative solution to problem F34a in [1] for the case of $F_2$. Two elements $u$ and $v$ in $F_n$ are said to be boundedly translation equivalent if the ratio of the cyclic lengths of $\phi(u)$ and $\phi(v)$ is bounded away from 0 and from $\infty$ for every automorphism $\phi$ of $F_n$. We provide an algorithm to determine whether or not two given elements of $F_2$ are boundedly translation equivalent, thus answering question F38c in the online version of [1] for the case of $F_2$. We further prove that there exists an algorithm to decide whether or not a given finitely generated subgroup of $F_2$ is the fixed point group of some automorphism of $F_2$, which settles problem F1b in [1] in the affirmative for the case of $F_2$.