Originally published in 1989, this is an advanced text and research monograph on groups acting on low-dimensional topological spaces, and for the most part the viewpoint is algebraic. Much of the book occurs at the one-dimensional level, where the topology becomes graph theory. Two-dimensional topics include the characterization of Poincare duality groups and accessibility of almost finitely presented groups. The main three-dimensional topics are the equivariant loop and sphere theorems. The prerequisites grow as the book progresses up the dimensions. A familiarity with group theory is sufficient background for at least the first third of the book, while the later chapters occasionally state without proof and then apply various facts which require knowledge of homological algebra and algebraic topology. This book is essential reading for anyone contemplating working in the subject.

This book offers a detailed introduction to graph theoretic methods in profinite groups and applications to abstract groups. It is the first to provide a comprehensive treatment of the subject. The author begins by carefully developing relevant notions in topology, profinite groups and homology, including free products of profinite groups, cohomological methods in profinite groups, and fixed points of automorphisms of free pro-p groups. The final part of the book is dedicated to applications of the profinite theory to abstract groups, with sections on finitely generated subgroups of free groups, separability conditions in free and amalgamated products, and algorithms in free groups and finite monoids. Profinite Graphs and Groups will appeal to students and researchers interested in profinite groups, geometric group theory, graphs and connections with the theory of formal languages. A complete reference on the subject, the book includes historical and bibliographical notes as well as a discussion of open questions and suggestions for further reading.

This outstanding new book presents the modern, geometric approach to group theory, in an accessible and engaging approach to the subject. Topics include group actions, the construction of Cayley graphs, and connections to formal language theory and geometry. Theorems are balanced by specific examples such as Baumslag-Solitar groups, the Lamplighter group and Thompson's group. Only exposure to undergraduate-level abstract algebra is presumed, and from that base the core techniques and theorems are developed and recent research is explored. Exercises and figures throughout the text encourage the development of geometric intuition. Ideal for advanced undergraduates looking to deepen their understanding of groups, this book will also be of interest to graduate students and researchers as a gentle introduction to geometric group theory.

In the last ?fteen years two seemingly unrelated problems, one in computer science and the other in measure theory, were solved by amazingly similar techniques from representation theory and from analytic number theory. One problem is the - plicit construction of expanding graphs («expanders»). These are highly connected sparse graphs whose existence can be easily demonstrated but whose explicit c- struction turns out to be a dif?cult task. Since expanders serve as basic building blocks for various distributed networks, an explicit construction is highly des- able. The other problem is one posed by Ruziewicz about seventy years ago and studied by Banach [Ba]. It asks whether the Lebesgue measure is the only ?nitely additive measure of total measure one, de?ned on the Lebesgue subsets of the n-dimensional sphere and invariant under all rotations. The two problems seem, at ?rst glance, totally unrelated. It is therefore so- what surprising that both problems were solved using similar methods: initially, Kazhdan’s property (T) from representation theory of semi-simple Lie groups was applied in both cases to achieve partial results, and later on, both problems were solved using the (proved) Ramanujan conjecture from the theory of automorphic forms. The fact that representation theory and automorphic forms have anything to do with these problems is a surprise and a hint as well that the two questions are strongly related.

For the first time, every finite group is represented in the form of a graph in this book. This study is significant because properties of groups can be immediately obtained by looking at the graphs of the groups.

From the reviews: "Béla Bollobás introductory course on graph theory deserves to be considered as a watershed in the development of this theory as a serious academic subject. ... The book has chapters on electrical networks, flows, connectivity and matchings, extremal problems, colouring, Ramsey theory, random graphs, and graphs and groups. Each chapter starts at a measured and gentle pace. Classical results are proved and new insight is provided, with the examples at the end of each chapter fully supplementing the text... Even so this allows an introduction not only to some of the deeper results but, more vitally, provides outlines of, and firm insights into, their proofs. Thus in an elementary text book, we gain an overall understanding of well-known standard results, and yet at the same time constant hints of, and guidelines into, the higher levels of the subject. It is this aspect of the book which should guarantee it a permanent place in the literature." #Bulletin of the London Mathematical Society#1

The field of topological graph theory has expanded greatly in the ten years since the first edition of this book appeared. The original nine chapters of this classic work have therefore been revised and updated. Six new chapters have been added, dealing with: voltage graphs, non-orientable imbeddings, block designs associated with graph imbeddings, hypergraph imbeddings, map automorphism groups and change ringing.Thirty-two new problems have been added to this new edition, so that there are now 181 in all; 22 of these have been designated as ``difficult'' and 9 as ``unsolved''. Three of the four unsolved problems from the first edition have been solved in the ten years between editions; they are now marked as ``difficult''.

The main theme of this book is the interplay between the behaviour of a class of stochastic processes (random walks) and discrete structure theory. The author considers Markov chains whose state space is equipped with the structure of an infinite, locally finite graph, or as a particular case, of a finitely generated group. The transition probabilities are assumed to be adapted to the underlying structure in some way that must be specified precisely in each case. From the probabilistic viewpoint, the question is what impact the particular type of structure has on various aspects of the behaviour of the random walk. Vice-versa, random walks may also be seen as useful tools for classifying, or at least describing the structure of graphs and groups. Links with spectral theory and discrete potential theory are also discussed. This book will be essential reading for all researchers working in stochastic process and related topics.

In 1937 there appeared a paper that was to have a profound influence on the progress of combinatorial enumeration, both in its theoretical and applied aspects. Entitled Kombinatorische Anzahlbest immungen jUr Gruppen, Graphen und chemische Verbindungen, it was published in Acta Mathematica, Vol. 68, pp. 145 to 254. Its author, George Polya, was already a mathematician of considerable stature, well-known for outstanding work in many branches of mathematics, particularly analysis. The paper in Question was unusual in that it depended almost entirely on a single theorem -- the "Hauptsatz" of Section 4 -- a theorem which gave a method for solving a general type of enumera tion problem. On the face of it, this is not something that one would expect to run to over 100 pages. Yet the range of the applica tions of the theorem and of its ramifications was enormous, as Polya clearly showed. In the various sections of his paper he explored many applications to the enumeration of graphs, principally trees, and of chemical isomers, using his theorem to present a comprehen sive and unified treatment of problems which had previously been solved, if at all, only by ad hoc methods. In the final section he investigated the asymptotic properties of these enumerational results, bringing to bear his formidable insight as an analyst