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

The pro-isomorphic zeta function ζΓ∧(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 Q_p-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 Δ _{m , 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 Δ _{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 2 m+ 3 is shown to be 6-15m+3.