Arithmetic and Zariski-dense subgroups in algebraic groups

该项目解决了半单代数群的算术稠密子群和一般Zariski稠密子群研究中的几个重要问题。该项目的中心主题之一是弱可公度Zariski-稠密子群的分析。弱可公度性的概念早在PI和G.Prasad的合作工作中被引入，它对算术群的分析导致了微分几何中涉及长度可公度和等谱算术定义的局部对称空间的几个问题的解决。在当前的项目中，一些以前只知道算术群的有限性结果有望推广到任意Zariski-稠密子群。这项工作可能会对非算术定义的局部对称空间产生有趣的结果。它还涉及到代数群论中的一个问题，即域K上的绝对几乎单代数群在多大程度上是由它的极大K-环面的同构类决定的。项目的这一部分将建立在PI分析具有相同极大子域的除法代数的最新结果的基础上。PI还打算继续研究同余子群问题。一般而言，该项目致力于基于矩阵群元素的特征值的信息来分析一类非常广泛的矩阵群(半单代数群的Zariski稠密子群)。这种方法在有限群的表示理论中是基本的，但正如PI和G.Prasad所发现的那样，它还可以用来刻画许多算术群(这些群是元素是具有整数矩阵的矩阵的特殊群)。该项目的目标是将其中一些结果推广到比算术更普遍的群体。这项工作适用于著名的问题“人们能听到鼓的形状吗？”在一些称为局部对称空间的特殊空间的背景下。

The project addresses several important problems in the investigation of arithmetic and general Zariski-dense subgroups of semisimple algebraic groups. One of the central themes in the project is the analysis of weakly commensurable Zariski-dense subgroups. The notion of weak commensurability was introduced earlier in a joint work of the PI and G. Prasad, and its analysis for arithmetic groups has led to a resolution of several problems in differential geometry dealing with length-commensurable and isospectral arithmetically defined locally symmetric spaces. In the current project, some finiteness results which were previously known only for arithmetic groups are expected to be generalized to arbitrary Zariski-dense subgroups. This work is likely to have interesting consequences for non-arithmetically defined locally symmetric spaces. It is also related to the problem in the theory of algebraic groups of to what extent an absolutely almost simple algebraic group over a field K is determined by the isomorphism classes of its maximal K-tori. This part of the project will build on the PI's recent results analyzing division algebras having the same maximal subfields. The PI also intends to continue the investigation of the congruence subgroup problem. Generally speaking, the project focuses on the analysis of a very broad class of matrix groups (Zariski-dense subgroups of semisimple algebraic groups) based on the information about the eigenvalues of their elements. This approach is fundamental in the representation theory of finite groups, but as was discovered by the PI and G. Prasad it can also be used to characterize many arithmetic groups (which are special groups whose elements are matrices with integer matrices). The goal of the project is to extend some of these results to groups much more general than arithmetic. This work has applications to the famous question ''Can one hear the shape of a drum?'' in the context of some special spaces called locally symmetric spaces.