Algorithmic problems for free-abelian times free groups

TL;DR

This paper investigates algorithmic problems in groups formed by direct products of free-abelian and free groups, providing explicit formulas and solutions for various decision problems.

Contribution

It introduces natural extensions of standard notions to these groups and offers explicit expressions for endomorphisms, solving key algorithmic and decision problems.

Findings

Solved membership, isomorphism, and finite index problems.

Provided explicit formulas for endomorphisms.

Addressed subgroup and coset intersection, fixed point, and Whitehead problems.

Abstract

We study direct products of free-abelian and free groups with special emphasis on algorithmic problems. After giving natural extensions of standard notions into that family, we find an explicit expression for an arbitrary endomorphism of $\ZZ^m \times F_n$. These tools are used to solve several algorithmic and decision problems for $\ZZ^m \times F_n $: the membership problem, the isomorphism problem, the finite index problem, the subgroup and coset intersection problems, the fixed point problem, and the Whitehead problem.