Fixed points and stable images of endomorphisms for the free group of rank two

TL;DR

This paper presents an algorithm to compute fixed subgroups and stable images of endomorphisms in the free group of rank two, addressing longstanding questions in group theory.

Contribution

It provides the first algorithmic solution for determining fixed points and stable images in $F_2$, solving questions posed by Stallings and Ventura.

Findings

Algorithm successfully computes fixed subgroups for any endomorphism of $F_2$.

Algorithm determines stable images, confirming theoretical predictions.

Answers longstanding open questions in the theory of free groups.

Abstract

We give an algorithm which computes the fixed subgroup and the stable image for any endomorphism of the free group of rank two $F_2$, answering for $F_2$ a question posed by Stallings in 1984 and a question of Ventura.