Subgroup growth.
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 group theory. It is amazing how many different areas of mathematics are involved in the subject: number theory, logic, analysis, probabilistic methods, combinatorics, algebraic groups, Lie algebras and of course all branches of group theory. Residually finite groups is a part of infinite group theory that enjoys a great amount of structural results from the second half of the last century. The most striking one is the positive solution of the Restricted Burnside Problem by Zelmanov. Subgroup growth can be considered as one of the approaches to bring order into residually finite groups. One can think of it as a measure of residual finiteness: fast subgroup growth means strong residual finiteness for a group and slow growth means weak residual finiteness. It can be even better expressed in terms of the profinite completion: fast subgroup growth implies a big profinite completion and slow growth implies a small one. So for profinite groups the interpretation is even more illuminating: subgroup growth somewhat estimates the size of a profinite group. For an arithmetic group the question about the size of its profinite completion is exactly the congruence subgroup problem. A small size of the profinite completion means the positive (or almost positive) solution of the congruence subgroup problem and a big size means that the solution is negative. This suggests a reformulation (and even a generalization) of the congruence subgroup problem in terms of subgroup growth. This is discussed in Chapter 7, one of the most interesting chapters of the book. Each chapter of the book deals with some particular aspect of subgroup growth. Chapter 2 considers free groups, the groups with fastest possible subgroup growth (\(n^n\)). Chapter 3 is concerned with groups of exponential growth which include in particular free pro-\(p\) groups of finite rank. The next chapter examines pro-\(p\) groups with growth smaller than exponential. It includes the pro-\(p\) PSG theorem that characterizes pro-\(p\) groups of polynomial growth: pro-\(p\) groups with this growth are pro-\(p\) analytic (or equivalently pro-\(p\) groups of finite rank). The other striking result is the ``gap theorem'' in the growth spectrum of pro-\(p\) growth: it says that the growth can not be faster than polynomial and slower than \(n^{c\log n}\), where \(c<1/(8\log p)\). The final section of the chapter is devoted to finitely presented pro-\(p\) groups; the main theorem here has Tits alternative flavour: it says that a finitely presented pro-\(p\) group either contains a free non-Abelian pro-\(p\) subgroup or has growth of type at most \(2^{\sqrt n}\). Chapter 5 is devoted to the main result of the book, the famous PSG theorem: a finitely generated residually finite group has polynomial subgroup growth if and only if it is virtually solvable of finite rank. Chapter 6 is concerned with counting congruence subgroups and Chapter 7 (already mentioned above) devoted to the congruence subgroup problem. Chapter 8 discusses stronger versions of the PSG for linear groups and residually nilpotent groups. In both cases there is a gap between polynomial growth and growth of type \(n^{\log n/\log\log n}\). Chapter 10 contains a characterization of profinite groups with polynomial subgroup growth. Comparing with the discrete situation, a profinite group with polynomial subgroup growth is not virtually prosolvable, but modulo a prosolvable normal subgroup of finite rank it is (virtually) a direct product of finite simple groups of Lie type with certain arithmetical conditions. The next Chapter employs probabilistic methods. One can consider a profinite group as a probability space using its natural Haar measure as a compact topological group (in \(G^k\)). Then the set of \(k\)-tuples generating a group \(G\) is measurable. If for some \(k\) the measure of that set is positive, one says that \(G\) is positively finitely generated. A theorem of Mann and Shalev relates this notion to the subject of the book. It states that a profinite group is positively finitely generated if and only if the growth of maximal subgroups of \(G\) is polynomial. Note that the `only if' part depends on the classification of finite simple groups. Chapter 12 relates subgroup growth with index growth \([G:G^n]\) and bounded generation. According to a celebrated result of Zelmanov if \(G\) is a finitely generated profinite (or residually finite discrete) group, then \([G:G^n]\) is finite and so one can consider its growth function. A group (profinite group) is said to have bounded generation if it is a product of its cyclic (procyclic) subgroups. The property plays an important role in the theory of arithmetic groups and the congruence subgroup problem. The main result of the section states that for a profinite group polynomial subgroup growth implies bounded generation and polynomial index growth. Chapter 13 is concerned with the question of completeness of the subgroup growth spectrum. It proves that for the class of profinite (or residually finite) groups the spectrum is essentially complete, i.e. unlike to the pro-\(p\) case, there is no gap in the growth spectrum. In the reviewer's opinion it would be very interesting to detect what gaps appear when one restricts to certain natural subclasses of profinite groups, like profinite groups of finite cohomological dimension (the discrete analog of the question is also very interesting). The Chapter ``Explicit formulas and asymptotics'' consists of interesting results but mostly without proofs. It contains explicit formulas of the subgroup growth for free products of finite or cyclic groups as well as surface groups. The last two Chapters are devoted to the \(\zeta\)-function \(\zeta_G(s)=\sum_{n=1}^\infty a_n(G)/n^s\). Historically, the topic started with the paper by Grunewald-Segal-Smith, and is developed by Grunewald and du Sautoy. In Chapter 15 the \(\zeta\)-function of nilpotent groups is studied. For these groups the \(\zeta\)-function has nice properties like an Euler product expansion, rationality for the local factor \(\zeta_{G,p}(s)=\sum_{n=1}^\infty a_{p^n}(G)p^{-n^s}\), analytic continuation. Finally Chapter 16 considers the \(\zeta\)-function of a compact \(p\)-adic analytic group. The local \(\zeta\)-function \(\zeta_{G,p}(s)=\sum_{n=1}^\infty a_{p^n}(G)p^{-n^s}\) is also rational for these groups, and in particular for the groups \(\text{SL}_n(\mathbb{Z}_p)\). Both Chapters give only an outline of the main ideas of the proofs. A wide range of topics are involved in the subject. To help the reader which is unfamiliar with one or another topic, the authors wrote 12 windows. The windows provide the necessary background, show the connections of the areas and interplay of the ideas. They are not just a list of results with references, but surveys written in accessible form to a non-specialist in the area and give the guidance and references to main results of the area. The book ends with open problems related to subgroup growth as a challenging invitation into the area! Both authors are excellent writers and The Ferran Sunyer i Balaguer Prize awarded for mathematical monographs of expository nature confirms this. This book is the first book in the area and contains many results that have not been published before. It is an important contribution to the mathematical literature. I am sure the book will have a remarkable impact and attract many young mathematicians to the subject.
scientific article; zbMATH DE number 1913818