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 parama 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.
Emmanuel Breuillard is a Professor at the Laboratoire de Mathematiques, Universite Paris-Sud Orsay. He was recently awarded an EMS Prize by the European Mathematical Society for his work in group theory. Hee Oh is a Professor of Mathematics at Brown University. She is an inaugural Fellow of the American Mathematical Society and has given a joint AMS-MAA invited address at the 2012 Joint Mathematics meetings and an invited lecture at the International Congress of Mathematicians in 2010.
Cambridge University Press
Date of Publication
Mathematical Sciences Research Institute Publications