Zeta Functions of Groups and Rings
Zeta functions have been a powerful tool in mathematics over the last two centuries. This book considers a new class of non-commutative zeta functions which encode the structure of the subgroup lattice in infinite groups. The book explores the analytic be
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Marcus du Sautoy Luke Woodward
Zeta Functions of Groups and Rings
1925
Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1925
Marcus du Sautoy · Luke Woodward
Zeta Functions of Groups and Rings
ABC
Marcus du Sautoy Luke Woodward Mathematical Institute University of Oxford 24-29 St Giles Oxford OX1 3LB, UK [email protected] [email protected]
ISBN 978-3-540-74701-7
e-ISBN 978-3-540-74776-5
DOI 10.1007/978-3-540-74776-5 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2007936935 Mathematics Subject Classification (2000): 20E07, 11M41 c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: design & production GmbH, Heidelberg Printed on acid-free paper 987654321 springer.com
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Preface
The study of the subgroup growth of infinite groups is an area of mathematical research that has grown rapidly since its inception at the Groups St. Andrews conference in 1985. It has become a rich theory requiring tools from and having 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 subgroup growth, in particular for torsion-free finitely-generated nilpotent groups. These zeta functions were introduced in [32], and other key papers in the development 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 significant 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 doctoral students of the first author which are yet to be made into printed form 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 s
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