## Discrete Subgroups of Semisimple Lie GroupsDiscrete subgroups have played a central role throughout the development of numerous mathematical disciplines. Discontinuous group actions and the study of fundamental regions are of utmost importance to modern geometry. Flows and dynamical systems on homogeneous spaces have found a wide range of applications, and of course number theory without discrete groups is unthinkable. This book, written by a master of the subject, is primarily devoted to discrete subgroups of finite covolume in semi-simple Lie groups. Since the notion of "Lie group" is sufficiently general, the author not only proves results in the classical geometry setting, but also obtains theorems of an algebraic nature, e.g. classification results on abstract homomorphisms of semi-simple algebraic groups over global fields. The treatise of course contains a presentation of the author's fundamental rigidity and arithmeticity theorems. The work in this monograph requires the language and basic results from fields such as algebraic groups, ergodic theory, the theory of unitary representatons, and the theory of amenable groups. The author develops the necessary material from these subjects; so that, while the book is of obvious importance for researchers working in related areas, it is essentially self-contained and therefore is also of great interest for advanced students. |

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### Contents

Introduction | 1 |

Preliminaries | 8 |

Density and Ergodicity Theorems | 79 |

Property T | 107 |

Factor Groups of Discrete Subgroups | 145 |

Characteristic Maps | 168 |

Discrete Subgroups and Boundary Theory | 195 |

Rigidity | 214 |

Normal Subgroups and Abstract Homomorphisms of Semisimple Algebraic Groups Over Global Fields | 258 |

Arithmeticity | 288 |

Appendices | 346 |

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### Common terms and phrases

action adjoint algebraic group applied arithmetic assertion assume automorphism Borel called central Chapter char characteristic closed closure cocycle commutative complete condition connected semisimple consequence consider contains continuous homomorphism Corollary deduce defined definition denote dense in H direct elements equality equivalent ergodic exists fact factor group fc-group field extension finite index follows function g e G given group G H(fc hand hence homomorphism identity induced infinite integer invariant irreducible isomorphism k-group lattice Lemma Let G Lie group linear locally compact matrix measure morphism natural normal subgroup observe obtain particular positive Proof Proposition prove rank G reductive relatively compact Remark replacing representation resp restriction satisfied sequence simply connected space subgroup F subset Suppose Theorem topological torus transformation unipotent uniquely unitary valuations variety vector Zariski dense

### Popular passages

Page 381 - On the congruence subgroup problem: determination of the "metaplectic kernel", Invent.

Page 385 - Volume preserving actions of lattices in semisimple groups on compact manifolds, Publ. Math. IHES, 59(1984), 5-33.

Page 381 - Raghunathan, MS A proof of Oseledec's multiplicative ergodic theorem. Israel J. Math.