Fixed subgroups and computation of auto-fixed closures in free-abelian times free groups

Abstract

The classical result by Dyer–Scott about fixed subgroups of finite order automorphisms of Fn being free factors of Fn is no longer true in Zm×Fn. Within this more general context, we prove a relaxed version in the spirit of Bestvina–Handel Theorem: the rank of fixed subgroups of finite order automorphisms is uniformly bounded in terms of m,n. We also study periodic points of endomorphisms of Zm×Fn, and give an algorithm to compute auto-fixed closures of finitely generated subgroups of Zm×Fn. On the way, we prove the analog of Day's Theorem for real elements in Zm×Fn, contributing a modest step into the project of doing so for any right angled Artin group (as McCool did with respect to Whitehead's Theorem in the free context).