The normal subgroup growth of a group captures the asymptotic behaviour of the lattice of its normal subgroups of finite index. The group's normal zeta function encodes the distribution of these subgroups quantiatively. Finitely generated nilpotent groups are arithmetic subgroups of unipotent algebraic groups. It therefore comes as no surprise that questions about their subgroup structure are best thought about, and answered, in number-theoretic and, occasionally, algebro-geometric terms.
I will explain a notion of uniformity for normal zeta functions, specifically for free nilpotent groups, and present a recent uniformity result for free nilpotent groups of class 2, answering a conjecture of Grunewald, Segal, and Smith in this case. My talk is based on recent joint work with Angela Carnevale and Michael Schein.