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 $$6-\frac{15}{m+3}$$ .