Profinite Groups and Residual Finiteness

This book describes the theory of profinite groups, from the basics of the theory to topics which are active areas of current research. It is the first textbook on profinite groups to make their use in studying residually finite groups via their profinite completions a central focus. The first part of the book gives the subject a firm theoretical underpinning from category theory and introduces profinite groups as objects worthy of study in their own right. The reader is not expected to have a background in category theory. The connection of a residually finite group to its profinite completion is explored in detail, with emphasis on various separability properties and profinite rigidity. The study of group cohomology is a key tool in the exploration of profinite groups. The central portion of this book gives a standalone first course in group cohomology before showing the modification of this theory for use with profinite groups. There is special emphasis on the unique features of profinite group cohomology, such as Pontryagin duality and Sylow theory. Later chapters of the book collect together for the first time important results concerning the relation of the cohomology of a group to that of its profinite completion and introduce the concept of an action of a profinite group on a profinite tree. This material aims to be a useful reference for researchers, as well as a learning resource.

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