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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.
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Subgroup Growth
is an extremely well-written book and is a delight to read. It has a wealth of information making a rich and timely contribution to an emerging area in the theory of groups which has come to be known as Asymptotic Group Theory. This monograph and the challenging open problems with which it concludes are bound to play a fundamental role in the development of the subject for many years to come.
Journal Indian Inst of Science
" [Subgroup growth] is one of the first books on Asymptotic Group Theory a new, quickly developing direction in modern mathematics The book of A. Lubotzky and D. Segal, leading specialists in group theory, answers questions in a beautiful way . It was natural to expect a text on the subject that would summarize the achievements in the field and we are very lucky to witness the appearance of this wonderful book. Readers will be impressed with the encyclopedic scope of the text. It includes all, or almost all, topics related to subgroup growth . The book also includes plenty of general information on topics that are well known to algebraic audiences and should be part of the background for every modern researcher in mathematics. a wonderful methodological tool introduced by the authors.
The book ends [with] a section on open problems, which contains 35 problems related to subgroup growth. The list will be useful and interesting to both established mathematicians and young researchers. There is no doubt that the list includes the most important and illuminating problems in the area, and we eagerly anticipate solutions of at least some of them in the near future.
The book will surely have [a] big impact on all readers interested in Group Theory, as well as in Algebra and Number Theory in general. "
Bulletin of the AMS
" The proofs in this book employ a remarkable variety of tools, from all branches of group theory, certainly, but also from number theory, logic, and analysis . The authors supply surveys, and some proofs, of necessary results, in the " windows" at the end of the book. These comprise about one quarter of the full book, and they give the needy reader a handy reference, without interrupting the flow of argument in the main text . Since the subject of this book is an active area of current research, there are many open problems in it . "
Mathematical Reviews
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