Discrete subgroups of semisimple Lie groups

本文讨论了半单李群的离散子群问题。这些问题是几种数学的相互作用。一方面，它们与有限域上的计数曲线有关，另一方面，它们与格的子群生长的渐近行为有关。PI已经解决了这方面的几个问题，并愿意给出一个更好的画面。该提议的第二个目的是描述具有给定覆盖空间的“最小”局部对称轨道，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如Siegel， Chinburg, Friedman, Meyerhoff, Gehring, Martin，和Lubotzky考虑过，并且在SL(2)中对不同局部域上的格进行了求解。对于大多数Chevalley群在一个正特征局部场上，PI解决了这个问题。PI想要了解这种轨道的结构。例如，他打算回答卢博茨基？关于最小的轨道是否紧凑的问题。下一个问题是Bruhat-Tits建筑上离散顶点传递行为的分类。自20世纪80年代以来，这些行动一直引起人们的兴趣。一些数学家试图构造这样的动作，到目前为止，对于大维度，只构造了一类这样的动作。它们也被用来构造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，有望有广泛的应用。私家侦探计划对这类行动进行分类。PI和Mohammadi在Bruhat-Tits构造的顶点集上构造了新的简单传递作用族，并给出了一个很强的分类定理，并且有可能得到正特征的新例子，从而得到新的Ramanujan复形。PI向Sarnak最近关于有限体积齐次空间中单幂元的素幂轨道的等分布猜想迈出了一步。半单群的离散子群可以看作是数学中若干部分的连接点。一方面，它们与几何和几何群论有关，另一方面，它们与动力系统和数论有关。PI就这一主题提出了若干相关问题。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量，或者用某些描述描述“最小”模型，这些描述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Möbius函数随机性的其他证据。这些项目包括不同的部分和不同的数学性质，如Bruhat-Tits理论，质量公式，子群增长和筛子理论。由于它与数学的广泛关系，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理解。

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.