On the degree of polynomial subgroup growth of nilpotent groups

Abstract

Let N be a finitely generated nilpotent group. The subgroup zeta function $\zeta _N^{\scriptscriptstyle \leqslant }(s)$ and the normal zeta function $\zeta _N^{\scriptscriptstyle \lhd }(s)$ of N are Dirichlet series enumerating the finite index subgroups or the finite index normal subgroups of N. We present results about their abscissae of convergence $\alpha _N^{\scriptscriptstyle \leqslant }$ and $\alpha _N^{\scriptscriptstyle \lhd }$, also known as the degrees of polynomial subgroup growth and polynomial normal subgroup growth of N, respectively. We first prove some upper bounds for the functions $N\mapsto \alpha _N^{\scriptscriptstyle \leqslant }$ and $N\mapsto \alpha _N^{\scriptscriptstyle \lhd }$ when restricted to the class of torsion-free nilpotent groups of a fixed Hirsch length. We then show that if two finitely generated nilpotent groups have isomorphic $\mathbb {C}$-Mal’cev completions, then their subgroup (resp. normal) zeta functions have the same abscissa of convergence. This follows, via the Mal’cev correspondence, from a similar result that we establish for zeta functions of rings. This result is obtained by proving that the abscissa of convergence of an Euler product of certain Igusa-type local zeta functions introduced by du Sautoy and Grunewald remains invariant under base change. We also apply this methodology to formulate and prove a version of our result about nilpotent groups for virtually nilpotent groups. As a side application of our result about zeta functions of rings, we present a result concerning the distribution of orders in number fields.