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

Mark N. Berman, Benjamin Klopsch, Uri Onn

Published 2015-11-23Version 1

The pro-isomorphic zeta function 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 $p$-adic 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; these groups can be viewed as generalizations 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 our groups and obtain, in particular, explicit formulae for the local pro-isomorphic zeta functions associated to $D^*$-groups of odd Hirsch length. The global abscissae of convergence of the pro-isomorphic zeta functions of $D^*$-groups of odd Hirsch length are determined.