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

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.