Zeta functions of groups and enumeration in Bruhat-Tits buildings

TL;DR

This paper presents a novel method for computing local normal zeta functions of certain nilpotent groups using vertex enumeration in Bruhat-Tits buildings, leading to explicit formulas and functional equations.

Contribution

It introduces a new approach based on Bruhat-Tits building enumeration for calculating local zeta functions of finitely generated torsion-free nilpotent groups.

Findings

Explicit formulas for class 2 groups with small centers

Derivation of local functional equations

Examples of non-uniform normal zeta functions

Abstract

We introduce a new method to calculate local normal zeta functions of finitely generated, torsion-free nilpotent groups. It is based on an enumeration of vertices in the Bruhat-Tits building for Sl_n(Q_p). It enables us to give explicit computations for groups of class 2 with small centres and to derive local functional equations. Examples include formulae for non-uniform normal zeta functions.