The workshop will focus on recent developments concerning various quantitative aspects of "thin groups". These are discrete subgroups of semisimple Lie groups which 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 (super-strong approximation).
Simultaneously and sometimes surprisingly, the study of thin groups turns out to be of fundamental importance in a variety of subjects, including equidistribution of homogeneous flows and lattice points counting problems, dynamics on Teichmuller space, the Bourgain-Gamburd-Sarnak sieve in orbit, and arithmetic or geometric properties of certain types of monodromy groups and coverings. The workshop will gather a variety of experts from group theory, number theory, ergodic theory and harmonic analysis to present the accomplishments to date to a broad audience and discuss directions for further study.
2-hour introductory mini-courses by:
1-hour talks by:
Notes by Peter Sarnak (PDF)
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Keywords and Mathematics Subject Classification (MSC)
Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification
No Secondary AMS MSC