Subgroup growth of free products
type: Item
instance of: scholarly article
title: Subgroup growth of free products
zbMATH Open document ID: 0862.20019
author: T. W. Müller
published in: Inventiones Mathematicae
publication date: 1997-05-13T00:00:00Z
review text: For a group \(G\) denote by \(s_G(n)\) the number of subgroups of index \(n\) in \(G\). The purpose of the present paper is to investigate the growth behaviour and the asymptotics of \(s_G(n)\) in the case when \(G\) is a free product of the form \(G=*^s_{k=1}G_k*F_r\) (\(0\leq r,s<\infty\), \(1<m_k=|G_k|<\infty\)). Here, the \(G_k\) are arbitrary finite groups and \(F_r\) denotes the free group of rank \(r\). The main result gives a complete asymptotic expansion for \(s_G(n)\) in terms of the numbers \(r\), \(s\), \(m_k\), \(s_{G_k}(d_k)\), \(d_k|m_k\). The author also considers the question as to what kind of information on this decomposition of a group \(G\) is contained in the function \(s_G(n)\) or its asymptotics, and he shows that this combinatorial information determines \(G\) up to finitely many isomorphism types.
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 11N45
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 931986
zbMATH Keywords: free products
zbMATH Keywords: subgroup growth
zbMATH Keywords: number of subgroups
zbMATH Keywords: asymptotic expansion
reviewed by: Alexander Ivanovich Budkin
MaRDI profile type: Publication
full work available at URL: s002220050091
OpenAlex ID: W2060491948
DOI: 10.1007/S002220050091
description: scientific article; zbMATH DE number 931986