Normal subgroup growth in nilpotent groups

The talk will report on joint work with Angela Carnevale and Christopher Voll and on work in progress with Tomer Bauer.

Let K be a number field with ring of integers O.  We explicitly determine the local factors, at all primes unramified in K, of the normal subgroup zeta functions of a large class of finitely generated class-2-nilpotent torsion-free groups over O.  This class includes the free class-2-nilpotent groups, various amalgamations of the Heisenberg group, and direct products of any these with abelian groups.  We study the analytic properties of these functions and also give some indication of what happens at the ramified primes.  In particular, these results unify and generalize work of many previous authors and prove a conjecture of Grunewald, Segal, and Smith from 1988 on the uniformity of normal zeta functions of finitely generated free class-2-nilpotent groups.