Intersection growth and zeta functions for nilpotent groups

Abstract
Intersection growth concerns the asymptotic behavior of the index $f(n)$ of the intersection of all subgroups of a group $G$ that have index $n$. In the case of f.g torsion free, nilpotent groups $G$ these numbers can be combined in a Dirichlet series and define a $\zeta$ function for the groups. Such zeta functions has many nice properties and are better behaved than the analogous function coming from subgroup growth or from representation growth.