On the intersection of fixed subgroups of Fn × Fm

We prove that, although it is undecidable if a subgroup fixed by an automorphism intersects nontrivially an arbitrary subgroup of [Formula: see text], there is an algorithm that, taking as input a monomorphism and an endomorphism of [Formula: see text], decides whether their fixed subgroups intersect nontrivially. The general case of this problem, where two arbitrary endomorphisms are given as input, remains unknown. We show that, when two endomorphisms of a certain type are given as input, this problem is equivalent to the Post Correspondence Problem for free groups.