A family of class-2 nilpotent groups, their automorphisms and pro-isomorphic zeta functions

TL;DR

This paper studies the pro-isomorphic zeta functions of a family of class-2 nilpotent groups, providing explicit formulas, functional equations, and analyzing their convergence properties and cluster points.

Contribution

It describes the automorphism groups of these nilpotent groups and computes their local and global pro-isomorphic zeta functions, extending known classifications.

Findings

Explicit formulas for local pro-isomorphic zeta functions.

Established functional equations and meromorphic continuations.

Identified cluster points in the spectrum of abscissae of convergence.

Abstract

The pro-isomorphic zeta function of a finitely generated nilpotent group $\Gamma$ is a Dirichlet generating function that enumerates finite-index subgroups whose profinite completion is isomorphic to that of $\Gamma$. Such zeta functions can be expressed as Euler products of $p$-adic integrals over the $p$-adic points of an algebraic automorphism group associated to $\Gamma$. In this way they are closely related to classical zeta functions of algebraic groups over local fields. We describe the algebraic automorphism groups for a natural family of class-$2$ nilpotent groups; these groups can be viewed as generalizations of $D^*$-groups of odd Hirsch length. General $D^*$-groups, that is `indecomposable' finitely generated, torsion-free class-$2$ nilpotent groups with central Hirsch length $2$, were classified up to commensurability by Grunewald and Segal. We calculate the local pro-isomorphic zeta functions for our groups and obtain, in particular, explicit formulae for the local pro-isomorphic zeta functions associated to $D^*$-groups of odd Hirsch length. From these we deduce local functional equations; for the global zeta functions we describe the abscissae of convergence and find meromorphic continuations. We deduce that the spectrum of abscissae of convergence for pro-isomorphic zeta functions of class-2 nilpotent groups contains infinitely many cluster points. For instance, the global abscissae of convergence of the pro-isomorphic zeta functions of $D^*$-groups of odd Hirsch length are determined and yield the cluster point $6$.