Mathematical Sciences: Discrete Subgroups of Lie Groups

数学科学：李群的离散子群

• 批准号: 9623256
• 负责人: Dave Witte
• 金额: $ 4.78万
• 依托单位: Oklahoma State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623256&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Discrete; Subgroups; Lie

Abstract Witte The project continues previous work of the PI on three related topics: superrigidity, tessellations of homogeneous spaces, and actions on the circle. 1) The Margulis Superrigidity Theorem applies to lattices only in semisimple groups, but the PI has extended this result to lattices in many other Lie groups. 2) The PI intends to continue investigating the structure of compact manifolds of the form D\G/H, where G is a simply connected Lie group and H and D are closed subgroups, such that D acts properly on G/H. One goal is to add to our understanding of which homogeneous spaes G/H admit such a tessellation D\G/H. 3) The PI proved that SL(3,Z) (and other arithmetic groups of higher Q-rank) has no interesting continuous actions on the circle or the real line. Attempts to extend this result to arithmetic groups whose Q-rank is 0 or 1 lead to interesting algebraic questions about arithmetic groups of higher R-rank: are they right orderable? are normal subgroups the only conjugation-invariant subsets that are closed under multiplication? Many important materials are crystals. The atomic structure of such a material is very symmetric, and a first step toward understanding it is to study the group formed by all the symmetries of the structure. This project investigates the symmetry groups that arise from crystals in mathematical spaces other than the 3-dimensional universe we live in. Basic problems are to understand which spaces do contain crystals, and when it can happen that crystals in two different spaces have the same group of symmetries.

摘要 Witte 该项目延续了 PI 之前在三个相关主题上的工作：超刚性、均匀空间的镶嵌以及圆上的作用。 1) Margulis 超刚性定理仅适用于半单群中的格，但 PI 已将此结果扩展到许多其他李群中的格。 2) PI打算继续研究D\G/H形式的紧流形结构，其中G是单连通李群，H和D是闭子群，使得D可以正确作用于G/H。 一个目标是加深我们对哪些齐次空间 G/H 允许这样的镶嵌 D\G/H 的理解。 3) PI 证明 SL(3,Z)（以及其他更高 Q 秩的算术群）在圆或实线上没有有趣的连续动作。 尝试将此结果扩展到 Q 秩为 0 或 1 的算术群会导致有关更高 R 秩算术群的有趣代数问题：它们是否可正确排序？ 正态子群是唯一在乘法下闭合的共轭不变子集吗？ 许多重要的材料都是晶体。 这种材料的原子结构非常对称，理解它的第一步是研究由结构的所有对称性形成的群。 该项目研究了我们生活的 3 维宇宙以外的数学空间中的晶体产生的对称群。基本问题是了解哪些空间确实包含晶体，以及两个不同空间中的晶体何时可能具有相同的对称群。