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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.
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