Introduction to Approximate Groups

Provides a comprehensive exploration of the main concepts and techniques from the young, exciting field of approximate groups.

Matthew C. H. Tointon (Author)

9781108470735, Cambridge University Press

Hardback, published 14 November 2019

216 pages, 3 b/w illus. 55 exercises
23.5 x 15.6 x 1.7 cm, 0.43 kg

'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, The Mathematical Gazette

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 The book also features a considerable amount of previously unpublished material, as well as numerous exercises and motivating examples.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.

1. Introduction
2. Basic concepts
3. Coset progressions and Bohr sets
4. Small doubling in abelian groups
5. Nilpotent groups, commutators and nilprogressions
6. Nilpotent approximate groups
7. Arbitrary approximate groups
8. Residually nilpotent approximate groups
9. Soluble approximate subgroups GLn(C)
10. Arbitrary approximate subgroups of GLn(C)
11. Applications to growth in groups
References
Index.

Subject Areas: Combinatorics & graph theory [PBV], Groups & group theory [PBG], Discrete mathematics [PBD]