Thin Groups and Superstrong Approximation

This collection of survey articles focuses on recent developments at the boundary between geometry, dynamical systems, number theory and combinatorics.

Emmanuel Breuillard (Edited by), Hee Oh (Edited by)

9781107036857, Cambridge University Press

Hardback, published 17 February 2014

376 pages
23.9 x 16.3 x 2.5 cm, 0.66 kg

This collection of survey and research articles focuses on recent developments concerning various quantitative aspects of 'thin groups'. There are discrete subgroups of semisimple Lie groups that are both big (i.e. Zariski dense) and small (i.e. of infinite co-volume). This dual nature leads to many intricate questions. Over the past few years, many new ideas and techniques, arising in particular from arithmetic combinatorics, have been involved in the study of such groups, leading, for instance, to far-reaching generalizations of the strong approximation theorem in which congruence quotients are shown to exhibit a spectral gap, referred to as superstrong approximation. This book provides a broad panorama of a very active field of mathematics at the boundary between geometry, dynamical systems, number theory and combinatorics. It is suitable for professional mathematicians and graduate students in mathematics interested in this fascinating area of research.

1. Some Diophantine applications of the theory of group expansion Jean Bourgain
2. A brief introduction to approximate groups Emmanuel Breuillard
3. Superstrong approximation for monodromy groups Jordan S. Ellenberg
4. The ubiquity of thin groups Elena Fuchs
5. The orbital circle method Alex V. Kontorovich
6. Sieve in discrete groups, especially sparse Emmanuel Kowalski
7. How random are word maps? Michael Larsen
8. Constructing thin groups Darren Long and Alan W. Reid
9. On ergodic properties of the Burger-Roblin measure Amir Mohammadi
10. Harmonic analysis, ergodic theory and counting for thin groups Hee Oh
11. Generic elements in Zariski-dense subgroups and isospectral locally symmetric spaces Gopal Prasad and Andrei Rapinchuk
12. Growth in linear groups László Pyber and Endre Szabó
13. On strong approximation for algebraic groups Andrei Rapinchuk
14. Generic phenomena in groups: some answers and many questions Igor Rivin
15. Affine sieve and expanders Alireza Salehi Golsefidy
16. Growth in linear groups Peter Sarnak.

Subject Areas: Number theory [PBH], Discrete mathematics [PBD], Mathematics [PB]