2015/11/19 by Mark Berman, Berman, Mark N., Benjamin Klopsch +3
Mathematics · #11M41 (Primary) 20E07 #14L40 (Secondary) #17B40 #17B45 #20F18 #20F69 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1511.06360
openalex publication_date 2015/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The pro-isomorphic zeta function of a finitely generated nilpotent group is a Dirichlet generating series that enumerates all finite-index subgroups whose profinite completion is isomorphic to that of the ambient group. We study the pro-isomorphic zeta functions of ℚ-indecomposable D^*-groups of even Hirsch length. These groups are building blocks of finitely generated class-two nilpotent groups with rank-two centre, up to commensurability. Due to a classification by Grunewald and Segal, they are parameterised by primary polynomials whose companion matrices define commutator relations for an explicit presentation. For Grunewald-Segal representatives of even Hirsch length of type f(t)=tm, we give a complete description of the algebraic automorphism groups of associated Lie lattices. Utilising the automorphism groups, we determine the local pro-isomorphic zeta functions of groups associated to t2 and t3. In both cases, the local zeta functions are uniform in the prime p and satisfy functional equations. The functional equations for these groups, not predicted by the currently available theory, prompt us to formulate a conjecture which prescribes, in particular, information about the symmetry factor appearing in local functional equations for pro-isomorphic zeta functions of nilpotent groups. Our description of the local zeta functions also yields information about the analytic properties of the corresponding global pro-isomorphic zeta functions. Some of our results for the D^*-groups associated to t2 and t3 generalise to two infinite families of class-two nilpotent groups that result naturally from the initial groups via `base extensions'.