On the degree of polynomial subgroup growth of nilpotent groups

TL;DR

This paper investigates the polynomial subgroup growth of finitely generated nilpotent groups through their zeta functions, establishing bounds, invariance under Mal'cev completions, and applications to ring zeta functions and number field orders.

Contribution

It introduces bounds for subgroup growth degrees, proves invariance of zeta function abscissae under Mal'cev isomorphisms, and extends results to virtually nilpotent groups and ring zeta functions.

Findings

Upper bounds for subgroup growth degrees in torsion-free nilpotent groups.

Invariance of zeta function abscissae under Mal'cev completions.

Application to distribution of orders in number fields.

Abstract

Let $N$ be a finitely generated nilpotent group. The subgroup zeta function $\zeta_N^{\leq}(s)$ and the normal zeta function $\zeta_N^\lhd(s)$ 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 $\alpha_N^\leq$ and $\alpha_N^\lhd$, 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 $N\mapsto \alpha_N^\leq$ and $N\mapsto\alpha_N^\lhd$ 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.