Groups, Graphs and Trees. An Introduction to the Geometry of Infinite Groups
This book provides an excellent introduction to geometric group theory. The exposition is divided into eleven chapters; those with odd numbers present general theory and those with even numbers analyse classical examples of infinite groups: groups generated by reflections, the Baumslag-Solitar group and the Thompson group. Apart from these, an example of a finitely generated infinite torsion group is constructed. The chapters providing general techniques are organized as follows. The first chapter summarizes results on group actions. The ”drawing trick” is explained in order to figure out some Cayley graphs. Finally, for groups acting on connected graphs, the notion of a fundamental domain is introduced. The third chapter presents free groups and free products from a geometric point of view. For example, it is shown that a group is free if and only if it acts freely on a tree. The Nielsen-Schreier theorem is then immediate. The fifth chapter gives information about Dehn's word problem and explains its connection to Cayley graphs. Chapter 7 demonstrates a use of automata and regular languages in group theory, including an interesting proof of the Howson theorem. In chapter 9, a finitely generated group with a fixed, finite set of generators is given the structure of a metric space. It is shown that any finitely generated group has a faithful representation as a group of isometries of a metric space. Various properties studied in previous chapters are intertwined with properties described by the metric. For example, it is proved that any almost convex group has a solvable word problem. The last chapter presents several properties of a Cayley graph of a finitely generated group that are independent of the choice of the finite set of generators. The book offers an interesting introductory course on the topic. Carefully chosen examples are an essential part of the exposition and they really help to understand general constructions. Apart from very elementary group theory, there are no other prerequisites. On the other hand, there are many suggestions for recent papers that use information contained in the book.