The Congruence Subgroups Problem and Groups of Finite Representation Type

这个建议继续调查的合理点群的简单代数群的数域和他们的S-算术子群。该项目的目标之一是证明新类群的同余核的中心性。 结合G.Prasad和A.Rapinchuk先前描述元核的结果，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Kazhdan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。这项研究是代数学与数论的交汇点。 线性群的正规子群结构的相关问题在上个世纪就有历史根源（Jordan，Klein），并且在本世纪一直是一个活跃的研究领域。 在过去10-15年的工具已经发展到研究一个中心问题的算术子群的代数群。这个问题，被称为同余子群问题，是与其他基本问题数论的新应用正在发现等领域的数据传输，数据处理和通信系统。

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.