Discrete subgroups of semisimple Lie groups

半单李群的离散子群

• 批准号: 1001598
• 负责人: Alireza Golsefidy
• 金额: $ 14.63万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2011-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001598&HistoricalAwards=false
• 关键词: Discrete; subgroups; semisimple; Lie; groups

This proposal addresses problems on discrete subgroups of semisimple Lie groups. These problems have been the interplay of several kinds of mathematics. They are, on one hand, related to counting curves over a finite field and, on the other, to the asymptotic behavior of the subgroup growth of lattices. The PI has already solved several problems in this area and would like to give a better picture. The second purpose of this proposal is to describe the "smallest" locally symmetric orbifolds with a given covering space, or its equivalent object in the non-Archimedean setting. This natural question had been considered by many mathematicians such as Siegel, Chinburg, Friedman, Meyerhoff, Gehring, Martin, and Lubotzky, and it is solved for lattices in SL(2) over different local fields. The PI solved this problem for most of Chevalley groups over a positive characteristic local field. The PI would like to understand structure of such orbifolds. For instance, he plans to answer Lubotzky?s question on whether the smallest orbifold is compact. The next problem is the classification of discrete vertex transitive actions on Bruhat-Tits buildings. These actions have been of interest since the 80?s. Several mathematicians have tried to construct such actions and, so far, for large dimensions, only one family of such actions have been constructed. They have been also used to construct explicit Ramanujan complexes. These combinatorial objects are generalization of Ramanujan graphs, which are highly useful in computer science, and expected to have broad applications. The PI plans to classify such actions. The PI and Mohammadi have constructed new families of simply-transitive actions on the vertex set of the Bruhat-Tits building and gave a very strong classification theorem, and it might be possible to get new examples in positive characteristic which in turn would give us new Ramanujan complexes. The PI proposes a step toward a recent conjecture by Sarnak on equi-distribution of orbits of prime powers of a unipotent element in a finite volume homogeneous space.Discrete subgroups of semi-simple groups can be considered as the connecting point of several parts of mathematics. On one hand, they are related to geometry and geometric group theory, and on the other to dynamical systems and number theory. The PI proposes several related problems on this subject. In these projects, PI would like to either count number of certain combinatorial objects with rich algebraic structure, or describe "smallest" models with certain descriptions, which usually have more symmetries and might be useful in computer science, or seek for other evidences of the randomness of the Möbius function. These projects consist of different parts and of various mathematical nature, e.g. Bruhat-Tits theory, mass formula, subgroup growth and sieve theory. Because of its relations with a wide range of mathematics, students at both graduate and under-graduate level can be exposed to and learn different topics. Moreover, some parts are of combinatorial or computational nature which makes them more accessible to under-graduates.

该提案解决了半单李群的离散子群的问题。这些问题是几种数学的相互作用。它们一方面与有限域上的计数曲线有关，另一方面与晶格子群增长的渐近行为有关。 PI 已经解决了该领域的几个问题，并希望提供更好的情况。该提案的第二个目的是描述具有给定覆盖空间的“最小”局部对称轨道折叠，或其在非阿基米德设置中的等效对象。这个自然问题已经被许多数学家考虑过，如西格尔、钦堡、弗里德曼、迈尔霍夫、格林、马丁和卢博茨基，并且它被解决了不同局部域上 SL(2) 中的格。 PI 通过积极的局部特征场为大多数 Chevalley 群体解决了这个问题。 PI 希望了解此类 Orbifolds 的结构。例如，他计划回答卢博茨基关于最小的环折是否紧凑的问题。下一个问题是 Bruhat-Tits 建筑物上离散顶点传递动作的分类。自 20 世纪 80 年代以来，这些行动一直备受关注。几位数学家试图构建这样的动作，但到目前为止，对于大维度，只构建了一系列这样的动作。它们还被用来构建显式拉马努金复合体。这些组合对象是拉马努金图的推广，在计算机科学中非常有用，并且有望具有广泛的应用。 PI 计划对此类行为进行分类。 PI 和 Mohammadi 在 Bruhat-Tits 大厦的顶点集上构建了新的简单传递作用族，并给出了非常强的分类定理，并且有可能获得正特征的新例子，这反过来会给我们新的拉马努金复合体。 PI 提出了 Sarnak 最近提出的关于有限体积齐次空间中单能元素素数幂轨道的均匀分布的猜想。半单群的离散子群可以被视为数学多个部分的连接点。一方面，它们与几何和几何群论有关，另一方面与动力系统和数论有关。 PI 就此主题提出了几个相关问题。在这些项目中，PI 希望要么计算某些具有丰富代数结构的组合对象的数量，要么用某些描述来描述“最小”模型，这些模型通常具有更多的对称性，可能在计算机科学中有用，或者寻找莫比乌斯函数随机性的其他证据。这些项目由不同的部分组成，具有不同的数学性质，例如Bruhat-Tits 理论、质量公式、子群增长和筛子理论。由于它与广泛的数学关系，研究生和本科生都可以接触和学习不同的主题。此外，某些部分具有组合或计算性质，这使得本科生更容易理解。