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…."