*Groups, Graphs and Trees* is an introduction to geometric group theory. Assuming some acquaintance with the fundamentals of group theory (e.g. direct products, Cayley’s theorem, free groups…), the author explains in the first chapter how one can attach, to any group, a graph on which the group acts naturally. Then, the structure of the group can be read on the structure of the corresponding graph.

The effectiveness of this point of view is clearly illustrated at many places throughout the book; geometric group theory provides clear and illustrative proofs of many difficult results (especially for finitely generated groups). It is a beautiful and modern approach to the study of infinite groups, and it deepens the understanding of many abstract algebraic problems. For instance, the natural action of a group on its Cayley graph is used to prove in chapter 3 that a group is free if and only if it acts freely on some tree, and as a corollary it is proved that every subgroup of a free group is free.

In order to keep the book at a reasonable length, and to make it accessible to a broader audience, the author as chosen to avoid any reference to abstract topology and to algebraic topology. Therefore, the point of view is mainly combinatorial (perhaps with the exception of chapter 9 where a metric approach, due to Gromov, is introduced). For the reader wishing to pursue the study of this theory, a comprehensive bibliography may serve as a guide to explore this vast and fascinating subject.

This beautiful book is accessible to advanced undergraduate students, having had a course on group theory; it is suitable for an introductory course on geometric group theory, and it will be surely of interest also to graduate students and to researchers who are not specialists in the field.

Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. He may be reached at mainardi2002@yahoo.fr.