Arithmetic groups, universal lattices and Kazhdan's property (T)

摘要奖项：DMS-0701639首席研究员：Yehuda Shalom该项目的研究方面围绕着两个主要相关的主题。第一个是关于任意环上线性群的Kazhdan性质。我们最近的结果是，在一定的条件下，n和环R，行列式一个n × n矩阵在R上有这个性质（任何等距作用在Hilbert空间上的不动点），涉及K-理论和分析工具的混合。它促进了关于这些群的性质和性质（T）的基本问题和推测。第二个主题解决了一个著名的问题齐默从1979年，对子群结构ofarithmetic集团，这已经触及到目前为止只有一个作者。 这个问题旨在完善马古利斯著名的正规子群定理，以了解几乎正规子群，即，它们的共轭相交于有限指数子群。 我们计划追求一种新的方法，它一方面会导致全新的结果和现象，即使是最简单的算术群体，另一方面也会导致许多有趣的自然问题。群论中关于有界生成的概念被证明与这个项目的两个方面都有很深的联系，尽管它来自不同的来源。群的数学概念是一般科学中最基本的概念之一，特别是在数学中，它使人们能够严格地对待和利用各种结构的对称性。 算术群是数学中一类特殊的群，它出现在许多数学领域，包括几何学、数论、代数学，甚至组合学和计算机科学中，并且是基本的。该项目的一个主要目的是揭示和研究这些群体的全新特征，这些特征以前在其他群体中也没有发现。另一个主要目标是首次系统地研究了更广泛的群体，概括了算术群体，并试图运用在过去的半个世纪里在更经典的环境中发展起来的一些丰富的理论和有力的工具。 这将有许多潜在的应用领域，算术组已经被证明是一个根本重要的对象和工具。

AbstractAward: DMS-0701639Principal Investigator: Yehuda ShalomThe research aspects of the project revolve around two mainrelated themes. The first pertains to Kazhdan's property oflinear groups over arbitrary finitely generated rings. Our recentresult that under certain conditions on n and the ring R, thegroup of determinant one n by n matrices over R has this property(fixed point for any isometric action on a Hilbert space),involves a mixture of K-theoretic and analytic tools. It promotesfundamental questions and speculations about both the nature ofthese groups, and property (T). The second topic addresses a wellknown question of Zimmer from 1979, on the subgroup structure ofarithmetic groups, which has been touched so far by only oneauthor. The question seeks refinement of Margulis' celebratednormal subgroup theorem, to the understanding of almost normalsubgroups, i.e., ones which intersect their conjugates in afinite index subgroup. We plan to pursue a new approach which onone hand leads to entirely new results and phenomena even for thesimplest arithmetic groups, and on the other to many intriguingnatural problems. The group theoretic notion of boundedgeneration turns out be deeply related to both aspects of theproject, although it arises from different sources.The mathematical notion of a group is one of the most fundamentalones in the sciences in general and in mathematics in particular,enabling one to rigorously treat as well as capitalize onsymmetries of various structures. A distinguished class ofgroups in mathematics is formed by the arithmetic groups, whichappear and are fundamental in many mathematical areas includinggeometry, number theory, algebra and even combinatorics andcomputer science. One main purpose of the project is to exposeand study entirely new features of these groups, which were notseen previously in other groups as well. Another main goal is tomake first systematic study of a wider family of groups,generalizing the arithmetic ones, and try to bring to bear someof the rich theory and powerful tools which have been developedover the past half a century in the more classical setting. Thiswould have many potential applications in those areas wherearithmetic groups have already proved to be a fundamentallyimportant object and tool.