Discrete subgroups of semisimple Lie groups

这个建议解决了关于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，另一方面与格子群增长的渐近行为有关。国际和平研究所已经解决了这方面的几个问题，并希望提供一个更好的图景。这一建议的第二个目的是描述具有给定覆盖空间的“最小”局部对称的或其在非阿基米德环境中的等价对象。西格尔、钦堡、弗里德曼、迈耶霍夫、盖林、马丁和卢博茨基等许多数学家都曾考虑过这个自然问题，并且解决了SL(2)中不同局部域上的格。PI在一个正特征局部域上解决了大多数Chvalley群的这个问题。PI想要了解这种ORBORBOLD的结构。例如，他计划回答卢博茨基和S关于最小的奥比福尔德是否紧凑的问题。下一个问题是Bruhat-Tits建筑物上离散顶点传递作用的分类。自上世纪80年代的S以来，这些作用就引起了人们的兴趣。几位数学家试图构造这种作用，但到目前为止，对于较大的维度，只有一族这样的作用被构造出来。它们还被用来建造显性的Ramanujan复合体。这些组合对象是Ramanujan图的推广，在计算机科学中有很高的实用价值，并有望有广泛的应用。PI计划对此类行为进行分类。Pi和Mohammadi在Bruhat-Tits建筑的顶点集上构造了新的单传递作用族，并给出了一个很强的分类定理，并且可能得到具有正特征的新例子，这反过来又给出了新的Ramanujan复形。PI向Sarnak最近提出的关于有限体积齐次空间中单位元素次方的轨道的均匀分布的猜想迈出了一步。半单群的离散子群可以被认为是数学的几个部分的连接点。它们一方面与几何和几何群论有关，另一方面与动力系统和数论有关。在这个问题上，PI提出了几个相关的问题。在这些项目中，PI想要计算具有丰富代数结构的某些组合对象的数目，或用某些描述来描述通常具有更多对称性且可能在计算机科学中有用的“最小”模型，或寻找Möbius函数随机性的其他证据。这些计划由不同的部分组成，具有不同的数学性质，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学科有关，研究生和本科生都可以接触和学习不同的主题。此外，有些部分是组合或计算性质的，这使得本科生更容易接触到它们。

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.