How Groups Grow

When studying groups, particularly finitely presented ones, a Cayley graph can be helpful to get a feeling for the general structure of the group. One observation that can be made using the Cayley graph is the general length of words in the group. Avinoam Mann’s How Groups Grow discusses growth functions, which describe the number of words of a given length within a fixed group. Through the course of the text the author gives, for several classes of groups, both explicit computations of and bounds for the growth function. He is careful to point out that these calculations are not always possible, a fact that leads into the major part of the text, which is on the classification and discussion of groups with certain types of growth functions, including polynomial, intermediate and exponential growth. In addition to these explicit algebraic computations manipulating word expressions within the groups, he also includes a description on how one would study and derive these bounds geometrically from the Cayley graph for these groups.

The book also contains the history of the study of growth, with references to the original papers. The author proves the majority of the theorems and provides the reader with the necessary group theoretic background. He is consistent in this throughout the text, making it a great resource for someone interested in learning about this topic. Often the author mentions that another proof for a theorem exists, and that it uses some theory that he has not yet developed. In every case, he opts to provide a (possibly less elegant) proof that the reader can fully understand, while providing references to allow the interested reader to look more deeply if necessary. Within the exposition the author includes many open problems that could serve to inspire the reader to work further in this field. The final chapter of the text contains a list of interesting open questions.

This is an excellent book and will serve as a wonderful reference. Though the author states that it was prepared from a series of lecture notes, it may not be the best choice for a textbook. There are some sections that include exercises of a reasonable level of difficulty, but the inclusion of exercises is not consistent throughout the book and the format of the sections is inconsistent. Several chapters contain no exercises, so if one were to use this in a course one would have to make up one’s own problems for these sections.

That being said, this book would be a nice supplement for a course on this topic. There appears to be no other book as entirely focused on the topic of growth. It is natural to compare this book to the only other book I could find with substantial coverage on this topic, Topics in Geometric Group Theory by Pierre de la Harpe. I found that Harpe’s text contained several sections devoted to the concept of growth, but the exposition was driven by examples and does not contain the complete proofs that Mann’s book contains. Mann’s text contains several topics not contained in Harpe’s book, including a complete proof of Gromov’s theorem which does not rely on Hilbert’s 5^th problem and asymptotic cones. The two books might complement each other well, providing excellent material for a course on this topic.

How Groups Grow is a complete reference on the topic of growth. After reading it one could confidently begin to study the growth of groups with a solid foundation of what is known in this field and the techniques that have been used in the past. It is an excellent introduction to anyone interested in beginning a research project in this area.

Ellen Ziliak is an Assistant Professor of mathematics at Benedictine University in Lisle IL. Her training is in computational group theory, particularly using geometric properties of groups. More recently she has become interested in ways to introduce undergraduate students to research in abstract algebra through applications.