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Zeta functions of groups and rings

2009-03-11

For a long time, zeta functions have been playing an exceptional role in propelling mathematical research and connecting seemingly different objects.

“The purpose of this stimulating book is to bring into print significant and as yet unpublished work from different areas of the theory of zeta functions of groups. There are numerous calculations of zeta functions of groups which are yet to be made into print. These explicit calculations provide evidence in favour of conjectures, or indeed can form inspiration and evidence for new conjectures”, summarizes Andrea Lucchini his review of Marcus du Sautoy's and Edward Woodward's “Zeta functions of groups and rings” [Lecture Notes in Mathematics 1925 (2008; Zbl 1151.11005)], and recalls:

“The study of the subgroup growth of infinite groups has grown rapidly since its inception at the Groups St. Andrews conference in 1985 [cf. Zbl 0596.00008]. It has become a rich theory with applications to many areas of group theory. Much of this progress is chronicled by A. Lubotzky and D. Segal within their book “Subgroup growth” [Basel: Birkhäuser (2003; Zbl 1071.20033)].

In this context, a natural development is the idea to study the “arithmetic of subgroup growth”, defining an object analogous to the Dedekind ζ-function of a number field: to the finitely generated group G the Dirichlet series

ζG (s)=∑ (an(G)/ns)

can be associated. The investigation of this function started with a paper by F. J. Grunewald, D. Segal and G. C. Smith [Invent. Math. 93, No. 1, 185-223 (1988; Zbl 0651.20040)] and the interest for this subject has grown explosively in the last few years.”

Then, the reviewer points out several important developments covered in this Lecture Notes, and concludes:

“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.”


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