Discrete Subgroups Generated by Lattices in Opposite Horospherical Subgroups

✍ Scribed by Hee Oh

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 582 KB
• Volume: 203
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1997.7188

✦ Synopsis

Contents. 0. Introduction. 1. Preliminaries. 1.1. Notation and terminology. 1.2. Some known algebraic lemmas. 1.3. Adjoint representation and maximal subgroups. 1.4. ‫-ޑ‬ forms of algebraic groups and ‫-ޑ‬rational representations. 1.5. Extension of ‫-ޑ‬forms. 2. The subgroups of the form ⌫ and ‫-ޑ‬forms. 2.1. Discrete subgroups in F , F 1 2 algebraic groups. 2.2. Generators of arithmetic groups. 2.3. Reflexive horospherical subgroups. 2.4. Margulis' theorem on representation the-Ž . ory and extension of ‫-ޑ‬forms. 2.5. The subgroup generated by Z U 1 Ž . and Z U . 2 3. Adjoint action on the space of lattices. 3.1. The space of lattices in algebraic unipotent groups. 3.2. Adjoint action. 3.3. Ratner's theorem on orbit closures. 3.4. Closedness of some orbits and ‫-ޑ‬forms. 4. The proof of the main theorem. 4.1. Commutative horospherical subgroup cases. 4.2. Heisenberg horospherical subgroup cases. 4.3. Non-‫-ޒ‬ Heisenberg horospherical subgroup cases. 4.4. Arithmetic subgroups of the form ⌫ . F , F 1 2 0. INTRODUCTION Let G be a center-free connected semisimple real algebraic group with no compact factors. The unipotent radical of a proper parabolic subgroup of G is called horospherical. Two horospherical subgroups are called opposite if they are the unipotent radicals of two opposite parabolic subgroups.