Universal factorization property of certain polycyclic groups

Let H be a torsion-free strongly polycyclic (torsion-free virtually polycyclic, resp.) group. Let G be any group with maximal condition. We show that there exists a torsion-free strongly polycyclic (torsion-free virtually polycyclic, resp.) group G and an epimorphism epsilon : G -> (G) over bar such that for any homomorphism (p : G -> H, it factors through (G) over tilde, i.e., there exists a homomorphism (phi) over tilde : G ->. H such that phi = (phi) over bar o epsilon. We show that this factorization property cannot be extended to any finitely generated group G. As an application of factorization, we give necessary and sufficient conditions for N(f, g) = R(f, g) to hold for maps f, g : X -> Y between closed orientable n-manifolds where pi(1) (X) has the maximal condition, Y is an infra-solvmanifold, N (f, g) and A(f, g) denote the Nielsen and Reidemeister coincidence numbers, respectively. (c) 2005 Elsevier B.V. All rights reserved.