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

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 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 - 2 n - 1 + m + n - 1 n - 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 - 15 m + 3 .