Computation of endo-fixed closures in free-abelian times free groups

TL;DR

This paper presents an algorithm to determine and compute fixed subgroups of endomorphisms in free-abelian times free groups, combining combinatorial and algebraic techniques.

Contribution

It introduces a novel algorithm that decides whether a subgroup is fixed by some endomorphisms and finds such endomorphisms if they exist.

Findings

Algorithm successfully determines fixed subgroups

Algorithm finds endomorphisms fixing the subgroup

Combines combinatorial and algebraic methods

Abstract

In this paper, we explore the behaviour of the fixed subgroups of endomorphisms of free-abelian times free (FATF) groups. We exhibit an algorithm which, given a finitely generated subgroup $\mathcal{H}$ of a FATF group $\mathcal{G}$, decides whether $\mathcal{H}$ is the fixed subgroup of some (finite) family of endomorphisms of $\mathcal{G}$ and, in the affirmative case, it finds such a family. The algorithm combines both combinatorial and algebraic methods.