On the minimal generator number of a polycyclic nilpotent-by-finite group

The determination of the minimal generator number d ( G ) of a polycyclic group G is one of the few still open problems in the algorithmic theory of these groups. If Ĝ G ^ is the profinite completion of G , then it is known that d(G) = d(Ĝ) d ( G ) = d ( G ^ ) or d(G) = d(Ĝ) +1 d ( G ) = d ( G ^ ) + 1 holds. The aim here is to introduce a construction for d(Ĝ) d ( G ^ ) if G is polycyclic and nilpotent-by-finite.