Subgroup Growth

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 Residuallynilpotent 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. - 2 Finite Simple Groups. - 3 Permutation Groups. - 4 Profinite Groups. - 5 Pro-p Groups. - 6 Soluble Groups. - 7 Linear Groups. - 8 Linearity Conditions for Infinite Groups. - 9 Strong Approximation for Linear Groups. - 10 Primes. - 11 Probability. - 12 p-adic Integrals and Logic. - Open Problems.