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Arithmeticity of discrete subgroups

Part of: Discontinuous groups and automorphic forms Lie groups Other groups of matrices

Published online by Cambridge University Press:  28 September 2020

YVES BENOIST
YVES BENOIST*
Affiliation:
CNRS, Université Paris-Sud, France (e-mail: [email protected])
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Abstract

The topic of this course is the discrete subgroups of semisimple Lie groups. We discuss a criterion that ensures that such a subgroup is arithmetic. This criterion is a joint work with Sébastien Miquel, which extends previous work of Selberg and Hee Oh and solves an old conjecture of Margulis. We focus on concrete examples like the group $\mathrm {SL}(d,{\mathbb {R}})$ and we explain how classical tools and new techniques enter the proof: the Auslander projection theorem, the Bruhat decomposition, the Mahler compactness criterion, the Borel density theorem, the Borel–Harish-Chandra finiteness theorem, the Howe–Moore mixing theorem, the Dani–Margulis recurrence theorem, the Raghunathan–Venkataramana finite-index subgroup theorem and so on.

Keywords

semisimple grouparithmetic groupparabolic grouplatticethin group

MSC classification

Primary: 22E40: Discrete subgroups of Lie groups
Secondary: 11F06: Structure of modular groups and generalizations; arithmetic groups 20H05: Unimodular groups, congruence subgroups
Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 9 , September 2021 , pp. 2561 - 2590
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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