This book is devoted to a study of the unit groups of orders in skew fields, finite dimensional and central over the rational field; it thereby belongs to the field of noncommutative arithmetic. Its purpose is a synopsis of results and methods, including full proofs of the most important results. It is addressed to researchers in number theory and arithmetic groups.
Inhaltsverzeichnis
0 Basic Facts. - 1 Hey s Theorem and Consequences. - 2 Siegel-Weyl Reduction Theory. - 3 The Tamagawa Number and the Volume of G(?)/G(?). - 3. 1 Statement of the main result. - 3. 2 Proof of 3. 1. - 3. 3 The volume of G(?)/G(?). - 4 The Size of ? . - 4. 1 Statement of results. - 4. 2 Proofs. - 5 Margulis Finiteness Theorem. - 5. 1 The Result. - 5. 2 Amenable groups. - 5. 3 Kazhdan s property (T). - 5. 4 Proof of 5. 1; beginning. - 5. 5 Interlude: parabolics and their opposites. - 5. 6 Continuation of the proof. - 5. 7 Contracting automorphisms and the Moore Ergodicity theorem. - 5. 8 End of proof. - 5. 9 Appendix on measure theory. - 6 A Zariski Dense and a Free Subgroup of ? . - 7 An Example. - 8 Problems. - 8. 1 Generators. - 8. 2 The congruence problem. - 8. 3 Betti numbers. - References.
