Approximate groups have shot to prominence in recent years, driven both by rapid progress in the field itself and by a varied and expanding range of applications. This text collects, for the first time in book form, the main concepts and techniques into a single, self-contained introduction. The author presents a number of recent developments in the field, including an exposition of his recent result classifying nilpotent approximate groups. The book also features a considerable amount of previously unpublished material, as well as numerous exercises and motivating examples. It closes with a substantial chapter on applications, including an exposition of Breuillard, Green and Tao's celebrated approximate-group proof of Gromov's theorem on groups of polynomial growth. Written by an author who is at the forefront of both researching and teaching this topic, this text will be useful to advanced students and to researchers working in approximate groups and related areas.

‘The book now under reviews offers an excellent introduction … the book is very nicely written, Researchers and fledgling researchers in this area will want to own this book.'

Mark Hunacek Source: The Mathematical Gazette

‘… an aspiring student who wants to enter the world of approximate groups will surely find the first chapters of the book, which cover the fundamentals, invaluable. Moreover, anyone willing to climb the mountain that is the BGT theorem should be grateful for the webbing ladders laid out in Chapters IV–VI. Less ambitious readers might still enjoy the small gems, scattered throughout the text, like Solymosi’s sum-product theorem in Chapter IX or the Sanders–Croot–Sisask power set argument in Chapter X, both of which are a delight to read… this is perhaps the first book that provides a systematic treatment of approximate groups as a mathematical subject. It is very likely to become one of standard texts in this rapidly developing field.’

Michael Bjorklund Source: Bulletin of the American Mathematical Society