Computing fixed closures in free groups

TL;DR

This paper introduces an algorithm to determine whether a subgroup of a free group is fixed by some automorphisms or endomorphisms, and constructs such a family if it exists, using combinatorial and geometric techniques.

Contribution

It provides the first decision procedure for fixed subgroups of free groups under automorphisms and endomorphisms, combining combinatorial and geometric methods.

Findings

Decides fixed subgroup property for subgroups of free groups

Constructs automorphisms or endomorphisms fixing the subgroup

Uses a novel combination of combinatorial and geometric approaches

Abstract

Let $F$ be a finitely generated free group. We present an algorithm such that, given a subgroup $H\leqslant F$, decides whether $H$ is the fixed subgroup of some family of automorphisms, or family of endomorphisms of $F$ and, in the affirmative case, finds such a family. The algorithm combines both combinatorial and geometric methods.