Generalized Igusa functions and ideal growth in nilpotent Lie rings

Carnevale, Angela; Schein, Michael M.; Voll, Christopher

doi:10.2140/ant.2024.18.537

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Generalized Igusa functions and ideal growth in nilpotent Lie rings

Angela Carnevale, Michael M. Schein and Christopher Voll

Vol. 18 (2024), No. 3, 537–582

DOI: 10.2140/ant.2024.18.537

Abstract

We introduce a new class of combinatorially defined rational functions and apply them to deduce explicit formulae for local ideal zeta functions associated to the members of a large class of nilpotent Lie rings which contains the free class-2-nilpotent Lie rings and is stable under direct products. Our results unify and generalize a substantial number of previous computations. We show that the new rational functions, and thus also the local zeta functions under consideration, enjoy a self-reciprocity property, expressed in terms of a functional equation upon inversion of variables. We establish a conjecture of Grunewald, Segal, and Smith on the uniformity of normal zeta functions of finitely generated free class-2-nilpotent groups.

Keywords

subgroup growth, ideal growth, normal zeta functions, ideal zeta functions, Igusa functions, combinatorial reciprocity theorems

Mathematical Subject Classification

Primary: 05A15, 11M41, 20E07

Milestones

Received: 13 June 2022

Revised: 28 February 2023

Accepted: 13 April 2023

Published: 16 February 2024

Authors

Angela Carnevale
School of Mathematical and Statistical Sciences University of Galway Galway Ireland

Michael M. Schein
Department of Mathematics Bar Ilan University Ramat Gan Israel

Christopher Voll
Faculty of Mathematics Bielefeld University Bielefeld Germany

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Vol. 18, No. 3, 2024