Factorization of group homomorphisms (Q1612121)

The authors show that if \(\pi\) is a (torsion-free) virtually polycyclic group, then for any finitely generated group \(G\) there exists a (torsion-free) virtually polycyclic group \(\overline G\) and an epimorphism \(\varepsilon_G\colon G\to\overline G\) such that every homomorphism \(\varphi\colon G\to\pi\) factors into \(\varphi=\overline\varphi\circ\varepsilon_G\) for some homomorphism \(\overline\varphi\colon\overline G\to\pi\). This unique factorization property is applied to obtain conditions for which the Nielsen coincidence number \(N(f,g)\) and the Reidemeister coincidence number \(R(f,g)\) coincide for maps \(f,g\colon X\to Y\) into an infra-solvmanifold \(Y\).