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

Abstract

The pro-isomorphic zeta function ζ∧Γ(s) of a finitely generated nilpotent group Γ is a Dirichlet generating function that enumerates finite-index subgroups whose profinite completion is isomorphic to that of Γ . Such zeta functions can be expressed as Euler products of p-adic integrals over the Qp -points of an algebraic automorphism group associated to Γ . 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 Δm,n of Hirsch length (m+n−2n−1)+(m+n−1n−1)+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 Δm,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 6−15m+3 . Show more