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

Abstract

The pro-isomorphic zeta function $\zeta ^\wedge _\Gamma (s)$ 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 $\mathbb {Q}_p$-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 $\Delta _{m,n}$ of Hirsch length $\left( {\begin{array}{c}m+n-2\\ n-1\end{array}}\right) +\left( {\begin{array}{c}m+n-1\\ n-1\end{array}}\right) + n$ and central Hirsch length n; these groups can be viewed as generalisations 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 the groups $\Delta _{m,n}$ 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 abscissa of convergence of the pro-isomorphic zeta function of a $D^*$-group of Hirsch length $2m+3$ is shown to be $6-\frac{15}{m+3}$.