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 Zm×FnZm×Fn. These tools are used to solve several algorithmic problems for Zm×FnZm×Fn: the membership problem, the isomorphism problem, the finite index problem, the subgroup and coset intersection problems, the fixed point problem, and the Whitehead problem.