Eventual centrality and Hopficity of direct products of finitely generated soluble groups

We prove that the direct product of two finitely generated soluble Hopfian groups is Hopfian, answering Kourovka Problem 5.38 affirmatively. For finitely generated soluble groups \(A,B\), every surjective endomorphism \(f\) of \(A\times B\) has a positive power \(F=f^m\) whose off-diagonal components satisfy \(F_{12}(B)\le Z(A)\) and \(F_{21}(A)\le Z(B)\). We also prove that a surjective endomorphism of \(A\times B\) with these centrality properties is injective whenever \(A,B\) are finitely generated Hopfian groups. This latter assertion requires neither solubility nor finite generation of the centers.