BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

The book under review is one of the first books on Asymptotic Group Theory—a new, quickly developing direction in modern mathematics which has links to many topics in Algebra, Analysis, Probability and Discrete Mathematics. Typically the subject of Asymptotic Group Theory is the study of the type of growth of various functions involving a natural parameter related to the group. Among the most important functions of this nature are the word growth and the subgroup growth functions, along with various modifications. While the word growth function γG(n) counts the number of elements of length no greater than n in the group, the subgroup growth function sG(n) counts the number of subgroups of index no greater than n in G. From a group-theoretical point of view, the natural questions are: 1. What are the general features of growth functions? 2. Which algebraic features of the group are reflected in the growth function? More broadly one might ask: 3. What are the applications of growth functions and what is their connection to other topics in mathematics and science? A number theorist might add to the above list: 4. What are the arithmetic properties of the sequences {sG(n)}n=1, {γG(n)}n=1, or any other growth function associated to a group G? In addition, Logicians, Computer Scientists and Geometers might ask questions that relate the growth to their own areas. The book of A. Lubotzky and D. Segal, leading specialists in group theory, answers these questions in a beautiful way in the context of subgroup growth. The subject was started about 20 years ago by the pioneering efforts and articles of F. Grunewald, A. Lubotzky, A. Mann, A. Shalev, D. Segal, and others, although some sporadic results, such as the recursive formula of M. Hall [Hal49] from 1949 for sF (n) in a free group F , appeared much earlier. During the last 20 years the subject of subgroup growth went through a period of tremendous development, as many mathematicians were involved in the process. As a result we now have a powerful theory, with new methods and tools, which has strong connections to many topics in Group Theory and Algebra in general, such as finite groups (including the classification of simple groups), nilpotent groups, solvable groups, groups acting on p-adic trees (including rooted trees), profinite groups, lattices in Lie groups and in algebraic groups, associative and Lie graded algebras, probability (especially random generation), number theory and p-adic analysis (zeta functions), and many others. After such a successful period, 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. Before describing the contents of this book,