Thin Groups and Superstrong Approximation [Hardcover]

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