Arithmetic Groups, Their Applications and Generalizations

该项目涉及算术群论和相关领域中的广泛问题。部分建议建立在PI与Gopal Prasad最近的工作基础上，其中引入了半单代数群的Zariski-稠密子群的弱可公度性的概念，并对其进行了分析，然后将其应用于微分几何中处理长度可公度和等谱局部对称空间的问题。这一领域的目标之一是完成绝对几乎单群的弱可公度算术子群的研究，并将结果推广到某些非算术群。进一步的目标包括研究一般半单群的子群和具有正特征的域上的群的弱可公度性。该提案讨论了这些结果在微分几何中的潜在应用。该方案的另一个重要组成部分是同余子群问题，对于与非对易除法代数相关的各向异性群来说，这个问题仍然没有解决。该方案讨论了同余核中心性的新准则，这些准则有望在新的情况下解决同余子群问题。此外，PI和G.Prasad计划写一本关于同余子群问题的书。与同余子群问题密切相关的是对代数群的有理点群的正规子群结构的研究；这里的目的之一是得到有限维除代数的乘法群的同余子群定理的最终形式，这将完成PI与Y.Segev和G.M.Seitz共同进行的一系列研究。该建议还包含了许多涉及算术群的各种推广的问题，从任意Zariski稠密子群到自由群的自同构群。算术群是特殊的群，其元素是具有整数项的矩阵。这个概念可以追溯到高斯关于积分二次型的工作，在数学的许多领域，包括代数和数论的各个部分(例如，自同构形式理论)，都发挥着至关重要的作用。近年来，算术群理论在代数和微分几何、李群和组合学中出现了新的应用。该提案侧重于算术群理论的几个重要方面以及潜在的应用。

The project addresses a broad range of problems in the theory of arithmetic groups and related areas. Part of the proposal builds on recent work of the PI with Gopal Prasad in which the notion of weak commensurability of Zariski-dense subgroups of semi-simple algebraic groups was introduced, analyzed and then applied to problems in differential geometry dealing with length-commensurable and isospectral locally symmetric spaces. One of the goals in this area is to complete the investigation of weakly commensurable arithmetic subgroups of absolutely almost simple groups and to extend the results to some nonarithmetic groups. Further goals include investigating weak commensurability for subgroups of general semi-simple groups and for groups over fields of positive characteristic. The proposal addresses potential applications of these results to differential geometry. Another important component of the proposal is the congruence subgroup problem, which remains unresolved for anisotropic groups associated with noncommutative division algebras. The proposal discusses new criteria for the centrality of the congruence kernel that are expected to lead to the resolution of the congruence subgroup problem in new cases. In addition, the PI and G. Prasad plan to write a book on the congruence subgroup problem. Closely related to the congruence subgroup problem is the investigation of the normal subgroup structure of the groups of rational points of algebraic groups; one of the objectives here is to obtain an ultimate form of the congruence subgroup theorem for the multiplicative group of a finite-dimensional division algebra, which would complete a long line of research conducted by the PI jointly with Y. Segev and G.M. Seitz. The proposal also contains a number of problems that involve various generalizations of arithmetic groups, ranging from arbitrary Zariski-dense subgroups to the automorphism groups of free groups. Arithmetic groups are special groups whose elements are matrices with integral entries. This notion, which can be traced back to the work of Gauss on integral quadratic forms, plays a crucial role in many areas of mathematics including algebra and various parts of number theory (e.g., the theory of automorphic forms). In recent years, new applications of the theory of arithmetic groups have emerged in algebraic and differential geometry, Lie groups and combinatorics. The proposal focuses on several important aspects of the theory of arithmetic groups as well as on potential applications.