Arithmetic and Zariski-dense subgroups in algebraic groups

该项目解决了几个重要的问题，在调查的算术和一般Zagliki稠密子群的半单代数群。 该项目的中心主题之一是分析弱可分解的Zagliki稠密子群。 在PI和G的联合工作中，较早地引入了弱可重复性的概念。Prasad和它对算术群的分析导致了微分几何中几个问题的解决，这些问题涉及可长度分解和等谱算术定义的局部对称空间。 在当前的项目中，一些有限性的结果，以前只知道算术群有望推广到任意Zurkiki稠密子群。 这项工作可能会有有趣的后果，非算术定义的局部对称空间。 它也与代数群理论中的问题有关，即域K上的绝对几乎单代数群在多大程度上由其最大K-环面的同构类决定。 这个项目的这一部分将建立在PI最近的结果分析师代数具有相同的最大子域。 PI还打算继续调查一致性子组问题。一般来说，该项目的重点是分析一类非常广泛的矩阵群（半单代数群的Zakirki稠密子群），基于有关其元素的特征值的信息。 这种方法是有限群表示论的基础，但正如PI和G。Prasad它也可以用来刻画许多算术群（这是特殊的群，其元素是具有整数矩阵的矩阵）。 该项目的目标是将其中一些结果扩展到比算术更一般的群体。 这项工作有应用到著名的问题“一个人能听到鼓的形状吗？在某些特殊空间的上下文中称为局部对称空间。

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.