Subgroup growth.
type: Item
instance of: scholarly article
title: Subgroup growth.
zbMATH Open document ID: 1071.20033
published in: Progress in Mathematics
publication date: 2003-05-20T00:00:00Z
review text: Subgroup growth, the main theme of the book under review, can be viewed as one of the topics of asymptotic group theory: the theory that studies the asymptotic behavior of various functions involving a natural parameter of the group. A well established example of such a function is the word-growth, a function that assigns to a natural number \(n\) the number of elements \(b_n^S(G)\) which can be expressed as a word of length at most \(n\) in some fixed set of generators \(S\) of the group \(G\). While the values \(b_n^S(G)\) depend on the choice of the generating set \(S\) of \(G\) its growth, i.e. the asymptotic behavior of the function \(b_n^S(G)\), does not and so it is an invariant of the group. Subgroup growth is a function that starts from the ``top'' of the group. It assigns to a natural number \(n\) the number of subgroups \(a_n(G)\) of \(G\) of index \(n\). Passing from the word-growth to subgroup growth we loose and gain. It is clear that in subgroup growth we loose the information that is below the intersection of all subgroups of finite index, so we study automatically residually finite groups. On the other hand, what we gain is a new area of group theory: profinite groups. The elements of profinite groups can not be expressed in general as words in generators, so there is no word growth for a profinite group. However, there is a one to one correspondence between subgroups of a given index of a group \(G\) and open subgroups of the same index of its profinite completion \(\widehat G\). This means that \(a_n(G)=a_n(\widehat G)\). In fact, subgroup growth is the feature of a profinite group, and can be viewed as one of the examples of the profinite group theory helping in studying abstract infinite groups. In practice, the authors move freely between profinite and abstract groups according to what is more appropriate to the context. No wonder, both authors have a taste for profinite groups: each of them has made a remarkable contribution in the profinite g…
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20-02
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20F69
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 1913818
zbMATH Keywords: subgroup growth
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: growth types
zbMATH Keywords: finitely generated groups
zbMATH Keywords: growth of congruence subgroups
zbMATH Keywords: subgroup-counting zeta functions
zbMATH Keywords: profinite groups
author: Alexander Lubotzky
author: Dan Segal
reviewed by: Pavel Zalesskij
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1913818