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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
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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