Arithmetic Groups and Tessellations of Homogeneous Spaces

算术群和齐次空间的镶嵌

• 批准号: 0100438
• 负责人: Dave Witte
• 金额: $ 8.74万
• 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: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100438&HistoricalAwards=false
• 关键词: Arithmetic; Groups; Tessellations; Homogeneous; Spaces

AbstractWitteOne focus of this project is the study of tessellations of homogeneous spaces. Namely, if G/H is a non-compact, simply connected homogeneous space of a connected Lie group G, the question is whether there is a properly discontinuous subgroup D of G, such that the orbit space D\G/H is compact. Some special cases were studied by L. Auslander, Y. Benoist, G. A. Margulis, R. J. Zimmer, and others. In collaboration with H. Oh and A. Iozzi, the PI has recently made progress in understanding the case where G is a semisimple Lie group of real rank two, including a detailed study of the case where G = SO(2,2n) or SU(2,2n). The PI will continue this research, both for real rank two and higher real rank. He will also continue his study of actions of arithmetic groups on the circle, and related questions.This project studies crystals in mathematical spaces other than the 3-dimensional universe that we live in. (A crystal is a material whose atomic structure is very symmetric.) The most fundamental problem in this subject is to decide which spaces contain crystals, and which do not. (For this question, the most interesting spaces are homogeneous, which means that every point of the space looks exactly like all of the other points.) Mathematicians have made substantial progress on this problem in recent years, and this project will continue the work. In cases where crystals do exist, the project will investigate the algebraic properties of the group formed by the symmetries of a crystal.

这个项目的一个重点是研究齐次空间的镶嵌。也就是说，如果G/H是连通李群G的非紧的、单连通的齐次空间，则问题是是否存在G的真不连续子群D，使得轨道空间D\G/H是紧的。L.Auslander，Y.Benoist，G.A.Marguis，R.J.Zimmer等人研究了一些特殊情况。最近，PI与H.oh和A.Iozzi合作，在理解G是实二阶半单李群的情况方面取得了进展，包括对G=SO(2，2n)或SU(2，2n)的情况进行了详细的研究。PI将继续这一研究，包括实数第二级和更高的实数列。他还将继续研究算术群在圆上的作用以及相关问题。这个项目研究的是数学空间中的晶体，而不是我们生活的三维宇宙中的晶体。(晶体是一种原子结构非常对称的材料。)这门学科最基本的问题是决定哪些空间含有晶体，哪些没有。(对于这个问题，最有趣的空间是均匀的，这意味着空间的每个点看起来都与所有其他点完全相同。)近年来，数学家们在这个问题上取得了实质性的进展，这个项目将继续这项工作。在晶体确实存在的情况下，该项目将研究由晶体的对称性形成的群的代数性质。