2021/09/09 by Diego Sulca, Sulca, Diego
Mathematics · #11M41 #20E07 #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2109.04580
openalex publication_date 2021/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let N be a finitely generated nilpotent group. The subgroup zeta function ζN≤(s) and the normal zeta function ζN^\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 αN^≤ and αN^\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↦ αN^≤ and N↦αN^\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 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.