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Birkhäuser - Birkhäuser Mathematics | Subgroup Growth

Subgroup Growth

Series: Progress in Mathematics, Vol. 212

Lubotzky, Alexander, Segal, Dan

2003, XXII, 454 p.

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Subgroup growth studies the distribution of subgroups of finite index in a group as a function of the index. In the last two decades this topic has developed into one of the most active areas of research in infinite group theory; this book is a systematic and comprehensive account of the substantial theory which has emerged.

As well as determining the range of possible "growth types", for finitely generated groups in general and for groups in particular classes such as linear groups, a main focus of the book is on the tight connection between the subgroup growth of a group and its algebraic structure. For example the so-called PSG Theorem, proved in Chapter 5, characterizes the groups of polynomial subgroup growth as those which are virtually soluble of finite rank. A key element in the proof is the growth of congruence subgroups in arithmetic groups, a new kind of "non-commutative arithmetic", with applications to the study of lattices in Lie groups. Another kind of non-commutative arithmetic arises with the introduction of subgroup-counting zeta functions; these fascinating and mysterious zeta functions have remarkable applications both to the "arithmetic of subgroup growth" and to the classification of finite p-groups.

A wide range of mathematical disciplines play a significant role in this work: as well as various aspects of infinite group theory, these include finite simple groups and permutation groups, profinite groups, arithmetic groups and strong approximation, algebraic and analytic number theory, probability, and p-adic model theory. Relevant aspects of such topics are explained in self-contained "windows", making the book accessible to a wide mathematical readership. The book concludes with over 60 challenging open problems that will stimulate further research in this rapidly growing subject.

Content Level » Research

Keywords » Abelian group - Algebra - Algebraic structure - Group theory - Prime - cls - number theory

Related subjects » Birkhäuser Mathematics

Table of contents 

0 Introduction and Overview.- 0.1 Preliminary comments and definitions.- 0.2 Overview of the chapters.- 0.3 On CFSG.- 0.4 The windows.- 0.5 The ‘notes’.- 1 Basic Techniques of Subgroup Counting.- 1.1 Permutation representations.- 1.2 Quotients and subgroups.- 1.3 Group extensions.- 1.4 Nilpotent and soluble groups.- 1.5 Abelian groups I.- 1.6 Finite p-groups.- 1.7 Sylow’s theorem.- 1.8 Rest riet ing to soluble subgroups.- 1.9 Applications of the ‘minimal index’.- 1.10 Abelian groups II.- 1.11 Growth types.- Notes.- 2 Free Groups.- 2.1 The subgroup growth of free groups.- 2.2 Subnormal subgroups.- 2.3 Counting d-generator finite groups.- Notes.- 3 Groups with Exponential Subgroup Growth.- 3.1 Upper bounds.- 3.2 Lower bounds.- 3.3 Free pro-p groups.- 3.4 Normal subgroups in free pro-p groups.- 3.5 Relations in p-groups and Lie algebras.- Notes.- 4 Pro-p Groups.- 4.1 Pro-p groups with polynomial subgroup growth.- 4.2 Pro-p groups with slow subgroup growth.- 4.3 The groups $$SL_r^1({\mathbb{F}_p}[[t]])$$.- 4.4 A-perfect groups.- 4.5 The Nottingham group.- 4.6 Finitely presented pro-p groups.- Notes.- 5 Finitely Generated Groups with Polynomial Subgroup Growth.- 5.1 Preliminary observations.- 5.2 Linear groups with PSG.- 5.3 Upper chief factors.- 5.4 Groups of prosoluble type.- 5.5 Groups of finite upper rank.- 5.6 The degree of polynomial subgroup growth.- Notes.- 6 Congruence Subgroups.- 6.1 The characteristic 0 case.- 6.2 The positive characteristic case.- 6.3 Perfect Lie algebras.- 6.4 Normal congruence subgroups.- Notes.- 7 The Generalized Congruence Subgroup Problem.- 7.1 The congruence subgroup problem.- 7.2 Subgroup growth of lattices.- 7.3 Counting hyperbolic manifolds.- Notes.- 8 Linear Groups.- 8.1 Subgroup growth, characteristic 0.- 8.2 Residually nilpotent groups.- 8.3 Subgroup growth, characteristic p.- 8.4 Normal subgroup growth.- Notes.- 9 Soluble Groups.- 9.1 Metabelian groups.- 9.2 Residually nilpotent groups.- 9.3 Some finitely presented metabelian groups.- 9.4 Normal subgroup growth in metabelian groups.- Notes.- 10 Profinite Groups with Polynomial Subgroup Growth.- 10.1 Upper rank.- 10.2 Profinite groups with wPSG: structure.- 10.3 Quasi-semisimple groups.- 10.4 Profinite groups with wPSG: characterization.- 10.5 Weak PSG = PSG.- Notes.- 11 Probabilistic Methods.- 11.1 The probability measure.- 11.2 Generation probabilities.- 11.3 Maximal subgroups.- 11.4 Further applications.- 11.5 Pro-p groups.- Notes.- 12 Other Growth Conditions.- 12.1 Rank and bounded generation.- 12.2 Adelic groups.- 12.3 The structure of finite linear groups.- 12.4 Composition factors.- 12.5 BG, PIG and subgroup growth.- 12.6 Residually nilpotent groups.- 12.7 Arithmetic groups and the CSP.- 12.8 Examples.- Notes.- 13 The Growth Spectrum.- 13.1 Products of alternating groups.- 13.2 Some finitely generated permutation groups.- 13.3 Some profinite groups with restricted composition factors.- 13.4 Automorphisms of rooted trees.- Notes.- 14 Explicit Formulas and Asymptotics.- 14.1 Free groups and the modular group.- 14.2 Free products of finite groups.- 14.3 Modular subgroup arithmetic.- 14.4 Surface groups.- Notes.- 15 Zeta Functions I: Nilpotent Groups.- 15.1 Local zeta functions as p-adic integrals.- 15.2 Alternative methods.- 15.3 The zeta function of a nilpotent group.- Notes.- 16 Zeta Functions II: p-adic Analytic Groups.- 16.1 Integration on pro-p groups.- 16.2 Counting subgroups in a p-adic analytic group.- 16.3 Counting orbits.- 16.4 Counting p-groups.- Notes.- Windows.- 1 Finite Group Theory.- 1 Hall subgroups and Sylow bases.- 2 Carter subgroups.- 3 The Fitting subgroup.- 4 The generalized Fitting subgroup.- 5 Tate’s theorem.- 6 Rank and p-rank.- 7 Schur multiplier.- 8 Powerful p-groups.- 9 GLn and Sym(n).- 2 Finite Simple Groups.- 1 The list.- 2 Generators.- 3 Subgroups.- 4 Representations.- 5 Automorphisms.- 6 Schur multipliers.- 7 An elementary proof.- 3 Permutation Groups.- 1 Primitive groups.- 2 Groups with restricted sections.- 3 Subgroups of alternating groups.- 4 Profinite Groups.- 1 Completions.- 2 Free profinite groups.- 3 Profinite presentations.- 5 Pro-p Groups.- 1 Generators and relations.- 2 Pro-p groups of finite rank.- 3 Linear pro-p groups over local fields.- 4 Automorphisms of finite p-groups.- 5 Hall’s enumeration principle.- 6 Soluble Groups.- 1 Nilpotent groups.- 2 Soluble groups of finite rank.- 3 Finitely generated metabelian groups.- 7 Linear Groups.- 1 Soluble groups.- 2 Jordan’s theorem.- 3 Monomial groups.- 4 Finitely generated groups.- 5 Lang’s theorem.- 8 Linearity Conditions for Infinite Groups.- 1 Variations on Mal’cev’s local theorem.- 2 Groups that are residually of bounded rank.- 3 Applications of Ado’s theorem.- 9 Strong Approximation for Linear Groups.- 1 A variant of the Strong Approximation Theorem.- 2 Subgroups of SLn(Fp).- 3 The ‘Lubotzky alternative’.- 4 Strong approximation in positive characteristic.- 10 Primes.- 1 The Prime Number Theorem.- 2 Arithmetic progressions and the Bombieri-Vinogradov theorem.- 3 Global fields and Chebotarev’s theorem.- 11 Probability.- 12 p-adic Integrals and Logic.- 1 Results.- 2 A peek inside the black box.- Open Problems.- 1 ‘Growth spectrum’.- 2 Normal subgroup growth in pro-p groups and metabelian groups.- 3 The degree of f.g. nilpotent groups.- 4 Finite extensions.- 5 Soluble groups.- 6 Isospectral groups.- 7 Congruence subgroups, lattices in Lie groups.- 8 Other growth conditions.- 9 Zeta functions.

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