The Congruence Subgroups Problem and Groups of Finite Representation Type

这一建议继续研究数域上简单代数群的有理点群及其S算术子群。该项目的目标之一是证明新的群类的同余核的中心性。结合G.Prasad和A.Rapinchuk等人描述亚可积核的结果，这将为这些群的同余子群问题提供一个明确的答案。PI还将研究有限生成群的有限维表示的刚性性质，以证明离散Kazhdan群具有有限表示类型。这项研究处于代数和数论的交汇点上。与线性群的正规子群结构有关的问题早在上个世纪(Jordan，Klein)就有了历史渊源，在本世纪一直是一个活跃的研究领域。在过去的10-15年里，研究代数群的算术子群的一个中心问题的工具已经被开发出来。这个问题被称为同余子群问题，它与数论中的其他基本问题有关，这些问题在数据传输、数据处理和通信系统等领域正在被发现新的应用。

This proposal continues investigations into groups of rational points of simple algebraic groups over number fields and of their S-arithmetic subgroups. One of the goals of the project is to prove the centrality of the congruence kernel for new classes of groups. In conjunction with the previous results of G.Prasad and A.Rapinchuk describing the metaplectic kernel, this will provide a definitive answer to the congruence subgroup problem for these groups. The PI will also study the rigidity property for finite dimensional representations of finitely generated groups in order to prove that discrete Kazhdan groups have finite representation type. This research lies in the meeting ground between algebra and number theory. Questions related to the normal subgroup structure of linear groups have historical roots in the previous century (Jordan, Klein), and have been an area of active research in this century. In the last 10-15 years tools have been developed to study a central problem about arithmetic subgroups of algebraic groups. This problem, known as the congruence subgroup problem, is connected with other fundamental problems in number theory for which new applications are being discovered in such areas as data transmission, data processing, and communication systems.