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

## TL;DR

This paper extends fixed subgroup theory to free-abelian times free groups, establishing bounds on fixed subgroup ranks, analyzing periodic points, and providing algorithms for auto-fixed closures, advancing understanding of automorphisms in this context.

## Contribution

It proves a bounded rank result for fixed subgroups, studies periodic points, and develops an algorithm for auto-fixed closures in Z^m×F_n, generalizing classical fixed subgroup results.

## Key findings

- Fixed subgroups have uniformly bounded rank in Z^m×F_n.
- An algorithm for computing auto-fixed closures is provided.
- Analog of Day's Theorem is established for real elements in Z^m×F_n.

## Abstract

The classical result by Dyer--Scott about fixed subgroups of finite order automorphisms of $F_n$ being free factors of $F_n$ is no longer true in $Z^m\times F_n$. 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 $Z^m\times F_n$, and give an algorithm to compute auto-fixed closures of finitely generated subgroups of $Z^m\times F_n$. On the way, we prove the analog of Day's Theorem for real elements in $Z^m\times F_n$, 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).