Algorithmic problems for free-abelian times free groups

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 ${\text{ZZ}}^m\times F_n$. These tools are used to solve several algorithmic and decision problems for ${\text{ZZ}}^m\times F_n$: the membership problem, the isomorphism problem, the finite index problem, the subgroup and coset intersection problems, the fixed point problem, and the Whitehead problem.