Zeta Functions of Groups and Rings

1. Front Matter

   Pages I-XII

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2. Introduction

   Pages 1-20

3. Nilpotent Groups: Explicit Examples

   Pages 21-68

4. Soluble Lie Rings

   Pages 69-82

5. Local Functional Equations

   Pages 83-119

6. Natural Boundaries I: Theory

   Pages 121-153

7. Natural Boundaries II: Algebraic Groups

   Pages 155-167

8. Natural Boundaries III: Nilpotent Groups

   Pages 169-177

9. Back Matter

   Pages 179-212

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Thestudyofthesubgroupgrowthofin?nitegroupsisanareaofmathematical research that has grown rapidly since its inception at the Groups St. Andrews conferencein1985.Ithasbecomearichtheoryrequiringtoolsfromandhaving applications to many areas of group theory. Indeed, much of this progress is chronicled by Lubotzky and Segal within their book [42]. However, one area within this study has grown explosively in the last few years. This is the study of the zeta functions of groups with polynomial s- groupgrowth,inparticularfortorsion-free?nitely-generatednilpotentgroups. These zeta functions were introduced in [32], and other key papers in the - velopment of this subject include [10, 17], with [19, 23, 15] as well as [42] presenting surveys of the area. The purpose of this book is to bring into print signi?cant and as yet unpublished work from three areas of the theory of zeta functions of groups. First, there are now numerous calculations of zeta functions of groups by doctoralstudentsofthe?rstauthorwhichareyettobemadeintoprintedform outside their theses. These explicit calculations provide evidence in favour of conjectures, or indeed can form inspiration and evidence for new conjectures. We record these zeta functions in Chap.2. In particular, we document the functional equations frequently satis?ed by the local factors. Explaining this phenomenon is, according to the ?rst author and Segal [23], “one of the most intriguing open problems in the area”.

• Lattice
• Zeta function
• algebra
• group
• ring

From the reviews:

"The book starts with a short lovely description of several classical zeta function … . It also contains a large number of examples of groups for which these zeta functions were explicitly computed. … it certainly will be a basic text for anyone who plans to work in this area. … These surely will be valuable for inspiring further developments." (Alexander Lubotzky, Mathematical Reviews, Issue 2009 d)

"The purpose of this stimulating book is to bring into print significant and as yet unpublished work from different areas of the theory of zeta functions of groups. … The book will be not only a valuable reference for people working in this area, but also a fascinating reading for everybody who wants to understand the role zeta functions have in group theory and the connections between subgroup growth and algebraic geometry over finite fields revealed by this theory." (Andrea Lucchini, Zentralblatt MATH, Vol. 1151, 2009)

“The authors have compiled a large body of facts and conjectures which will no doubt be most valuable for everyone working in this fascinating and very active field of research.” (C. Baxa, Monatshefte für Mathematik, Vol. 160 (3), June, 2010)

• Mathematical Institute, University of Oxford, Oxford, UK

  Marcus du Sautoy, Luke Woodward

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