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Zbl 1151.11005
du Sautoy, Marcus; Woodward, Luke
Zeta functions of groups and rings. (English)
[B] Lecture Notes in Mathematics 1925. Berlin: Springer. xii, 208~p. EUR~39.95/net; SFR~70.00; \$~59.95; \sterling~30.50 (2008). ISBN 978-3-540-74701-7/pbk

Let $G$ be a finitely generated group: a sequence of non-negative integers can be associated with $G,$ considering, for any $n \in \Bbb N,$ the number $a_G(n)$ of subgroups with index $n$. By the ``subgroup growth" of $G$ one means the asymptotic behaviour of this sequence. 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 {\it A. Lubotzky} and {\it 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 $\zeta$-function of a number field: to the finitely generated group $G$ the Dirichlet series $\zeta_G(s) = \sum \frac {a_n(G)}{n^s}$ can be associated. The investigation of this function started with a paper by {\it F. J. Grunewald, D. Segal} and {\it 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. The subgroup zeta function $\zeta_G(s)$ (and its variant $\zeta_G^\vartriangleleft(s)$ defined considering only the normal subgroups of $G$) has been deeply studied in the particular case of torsion-free finitely generated nilpotent groups ({$\Cal T$}-groups). In that case a useful tool is the Mal'cev correspondence between $\Cal T$-groups and nilpotent Lie rings. This allows to relate the subgroup zeta function of a $\Cal T$-group $G$ with the subring zeta function of the associated Lie ring $L$ (and in a similar way to relate the normal zeta function of $G$ with the ideal zeta function of $L).$ The purpose of this stimulating book is to bring into print significant and as yet unpublished work from different areas of the theory of zeta functions of groups.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. These zeta functions are recorded in Chapter 2, with particular emphasis on functional equations satisfied by the local factors. A significant discovery was a group where all but perhaps finitely many of the local zeta functions counting normal subgroups do not possess such a functional equation. Prior to this discovery, it was expected that all zeta functions of groups should satisfy a functional equations. Prompted by this counterexample, a conjecture has been outlined which offers a substantial demystification of this phenomenon. This conjecture and its ramifications are discussed in Chapter 4. Several years before the paper by Grunewald, Segal and Smith, the idea of associating a zeta function to a group was already introduced by Hey (and subsequently developed by other authors) in connection with the study of algebraic groups. Grunewald, Segal and Smith noticed that zeta functions of algebraic groups were in fact counting subgroups in nilpotent groups, thus extending Hey's original motivation for the investigation of these functions. The book also includes results in this direction. In a joint paper of {\it F. Grunewald} and the second author [C. R. Acad. Sci. Paris, Sér. I, Math. 327, No. 1, 1--6 (1998; Zbl 0986.11062)] it was announced that the zeta functions of algebraic groups of types $B_l,$ $C_l$ and $D_l$ all possess a natural boundary, but this work is also yet to be made into print. In Chapter 5 the authors present a theory of natural boundaries of two-variable polynomials. This is followed by Chapter 6 where the aforementioned result on the zeta functions of classical groups is proved, and Chapter 7, where the natural boundaries are considered of the zeta functions attached to nilpotent groups listed in Chapter 2. 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 (Padova)]

MSC 2000:: *11-02 Research monographs (number theory)
11M41 Other Dirichlet series and zeta functions
11S40 Zeta functions and L-functions of local number fields
14G10 Zeta-functions and related questions
20E07 Subgroup theorems (group theory)
20G30 Linear algebraic groups over global fields and their integers

Keywords: zeta functions of groups; finitely generated nilpotent groups; arithmetic of nilpotent groups; subgroup growth; Lie algebra zeta function; cone integral; ghost zeta function; Euler product of uniformly rational functions; meromorphic functions

Citations: Zbl 1071.20033; Zbl 0651.20040; Zbl 0986.11062; Zbl 0596.00008

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