Discrete subgroups of semisimple Lie groups

这个建议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，另一方面与格的子群增长的渐近行为有关。PI已经解决了这方面的几个问题，并希望提供更好的图片。这个建议的第二个目的是描述“最小”局部对称orbifolds与一个给定的覆盖空间，或其等价的对象在非阿基米德设置。许多数学家，如Siegel、Chinburg、Friedman、Meyerhoff、Gehring、Martin和Lubotzky，都考虑过这个自然问题，并且在SL（2）中的不同局部域上解决了这个问题。PI解决了这个问题，大多数Chevalley组在一个积极的特征局部领域。主要研究者希望了解此类眼眶的结构。例如，他计划回答Lubotzky？最小的orbifold是否是紧的。下一个问题是Bruhat-Tits建筑物上离散顶点传递作用的分类。这些行动自80年代以来一直受到关注。S.一些数学家试图构造这样的作用量，到目前为止，对于大维数，只有一族这样的作用量被构造出来。它们也被用来构建明确的拉马努金复合物。这些组合对象是Ramanujan图的推广，Ramanujan图在计算机科学中非常有用，并有望有广泛的应用。PI计划对此类行动进行分类。PI和Mohammadi在Bruhat-Tits建筑的顶点集上构造了新的简单传递作用族，并给出了一个非常强的分类定理，并且有可能得到正特征的新例子，这反过来又会给我们新的Ramanujan复形。PI向Sarnak最近提出的关于有限体积齐性空间中幂幺元的素数幂的轨道等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几个部分的连接点。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就这一问题提出了几个相关问题。在这些项目中，PI希望计算具有丰富代数结构的某些组合对象的数量，或者用某些描述来描述“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。这些项目包括不同的部分和各种数学性质，例如Bruhat-Tits理论，质量公式，子群增长和筛理论。由于它与广泛的数学关系，研究生和本科生水平的学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的性质，使他们更容易接触到本科生。

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.