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

[摘要]获奖：dms -0701639首席研究员：Yehuda shalom该项目的研究方面围绕两个主要相关主题。第一个是关于任意有限生成环上线性群的Kazhdan性质。我们最近的结果是，在n和环R上的某些条件下，R上的行列式1 n × n矩阵群具有这个性质（希尔伯特空间上任何等距作用的不动点），涉及k理论和分析工具的混合。它提出了关于这些群体的性质和性质(T)的基本问题和猜测。第二个主题解决了齐默1979年提出的关于算术群的子群结构的一个著名问题，迄今为止只有一位作者触及过这个问题。这个问题寻求对马古利斯著名的正规子群定理的改进，以理解几乎正规的子群，即在有限指数子群中相交的共轭子群。我们计划寻求一种新的方法，一方面可以为最简单的算术群带来全新的结果和现象，另一方面可以解决许多有趣的自然问题。有界生成的群论概念与项目的两个方面都有深刻的关系，尽管它的来源不同。群的数学概念是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非对称性。算术群是数学中一个特殊的群体，它在许多数学领域都很重要，包括几何、数论、代数，甚至组合学和计算机科学。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。另一个主要目标是对更广泛的群族进行第一次系统的研究，推广算术群，并尝试将过去半个世纪以来在更经典的背景下发展起来的一些丰富的理论和强大的工具引入其中。这将有许多潜在的应用领域，其中算术组已经被证明是一个重要的对象和工具。

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.