I have a torsion-free non-abelian nilpotent group $\Gamma$ of cohomological dimension $n$. Is it possible to say anything about the number of generators of $\Gamma$ in a minimal presentation?


Can I assume that the number of generators can be chosen to be less than $n$?



Cohomological dimension of groups & number of generators
Finitely generated torsion-free nilpotent groups are polycyclic. Therefore, their cohomological dimension equals their Hirsch length.This is a result of Gruenberg. One can find it in Gruenberg's book 'Cohomological topics in group theory' in section 8.8 or in Robert Bieri's Book on 'homological dimension of discrete groups' as Th. 7.14.


On the other hand, the minimal number of generators of a polycyclic group is smaller or equal than its Hirsch length. (This is proven by a simple induction argument on the Hirsch length)


