Ulteriori informazioni
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.
Sommario
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.
Riassunto
This collection of survey and research articles focuses on recent developments in quantitative aspects of 'thin groups'. It provides a broad panorama of a very active field at the boundary between geometry, dynamical systems, number theory and combinatorics for professional mathematicians and graduate students in this fascinating area of research.
