Rational and transcendental growth series for the higher Heisenberg groups
type: Item
instance of: scholarly article
title: Rational and transcendental growth series for the higher Heisenberg groups
zbMATH Open document ID: 0869.20018
published in: Inventiones Mathematicae
publication date: 1997-06-22T00:00:00Z
review text: Let \(G\) be a 2-step nilpotent group whose commutator subgroup \([G,G]\) is infinite cyclic. Every such group has a subgroup of finite index of the form \(H_n\times Z^m\), where \[ \begin{multlined} H_n=\langle x_1,\dots,x_n,y_1,\dots,y_n,z\mid[x_i,x_j]=[y_i,y_j]=1,\\ [x_i,y_j]=1\text{ for }i\neq j,\quad[x_i,y_i]=z,\quad[z,x_i]=[z,y_i]=1\rangle\end{multlined} \] is the \(n\)-th discrete Heisenberg group. We call \(n\) the Heisenberg rank of \(G\). In the paper under review the author proves that \(G\) has a finite generating set \(S\) such that \(G\) has rational growth with respect to \(S\). On the other hand, he shows that if \(G\) has Heisenberg rank at least two, then \(G\) has a finite generating set \(S\) such that \(G\) has transcendental growth with respect to \(S\). In particular, all higher discrete Heisenberg groups \(H_2,H_3,\dots\) possess transcendental growth series. Moreover the second Heisenberg group \(H_2\) has transcendental growth series with respect to its standard generating set \(\Sigma_2=\{x^{\pm1}_1,x^{\pm1}_2,y^{\pm1}_1,y^{\pm1}_2\}\).
reviewed by: N. Yu. Makarenko
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 931985
zbMATH Keywords: nilpotent groups
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: discrete Heisenberg groups
zbMATH Keywords: Heisenberg rank
zbMATH Keywords: finite generating sets
zbMATH Keywords: rational growth
zbMATH Keywords: transcendental growth series
author: Michael Stoll
MaRDI profile type: Publication
full work available at URL: s002220050090
OpenAlex ID: W2068140781
DOI: 10.1007/S002220050090
description: scientific article; zbMATH DE number 931985