Subgroup Growth

  • Alexander Lubotzky
  • Dan Segal
Part of the Progress in Mathematics book series (PM, volume 212)

Table of contents

  1. Front Matter
    Pages i-xxii
  2. Introduction and Overview

    1. Alexander Lubotzky, Dan Segal
      Pages 1-9
    2. Alexander Lubotzky, Dan Segal
      Pages 11-36
    3. Alexander Lubotzky, Dan Segal
      Pages 37-50
    4. Alexander Lubotzky, Dan Segal
      Pages 51-72
    5. Alexander Lubotzky, Dan Segal
      Pages 73-90
    6. Alexander Lubotzky, Dan Segal
      Pages 91-109
    7. Alexander Lubotzky, Dan Segal
      Pages 111-132
    8. Alexander Lubotzky, Dan Segal
      Pages 133-152
    9. Alexander Lubotzky, Dan Segal
      Pages 153-160
    10. Alexander Lubotzky, Dan Segal
      Pages 161-175
    11. Alexander Lubotzky, Dan Segal
      Pages 177-200
    12. Alexander Lubotzky, Dan Segal
      Pages 201-217
    13. Alexander Lubotzky, Dan Segal
      Pages 219-242
    14. Alexander Lubotzky, Dan Segal
      Pages 243-267
    15. Alexander Lubotzky, Dan Segal
      Pages 269-284
    16. Alexander Lubotzky, Dan Segal
      Pages 285-308
    17. Alexander Lubotzky, Dan Segal
      Pages 309-318
  3. Windows

    1. Alexander Lubotzky, Dan Segal
      Pages 319-327
    2. Alexander Lubotzky, Dan Segal
      Pages 329-336

About this book

Introduction

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.

Keywords

Abelian group Algebra Algebraic structure Group theory Prime cls number theory

Authors and affiliations

  • Alexander Lubotzky
    • 1
  • Dan Segal
    • 2
  1. 1.Institute of MathematicsHebrew UniversityJerusalemIsrael
  2. 2.Mathematical InstituteAll Souls CollegeOxfordUK

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-0348-8965-0
  • Copyright Information Birkhäuser Basel 2003
  • Publisher Name Birkhäuser Basel
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-0348-9846-1
  • Online ISBN 978-3-0348-8965-0
  • Series Print ISSN 0743-1643
  • Series Online ISSN 2296-505X
  • About this book