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.
The results of the project contribute to various aspects of the theory of arithmetic groups and its applications, particularly to differential geometry. Arithmetic groups are special groups whose elements are matrices with integer entries. This notion can be traced back to the work of Gauss on integral quadratic forms. Later, they played a prominent role in the development of the classical theory of automorphic forms. The foundations of the theory of arithmetic groups were laid by Harish-Chandra, Borel and Serre in the 60s. Since then, different aspects of the theory of arithmetic groups have been the subject of very active research marked by rich connections to other areas of mathematics such as number theory, combinatorics and geometry and topology. The results of the project contribute, in particular, to the investigation of the famous question in differential geometry ``Can one hear the shape of a drum?'' The previous work of G.Prasad and the PI reduced this question for locally symmetric spaces to the analysis of a new relationship between arithmetic groups called weak commensurability. This relationship provides an efficient way of matching the eigenvalues of elements of two arithmetic groups, which may even be represented by matrices of different sizes. So, the investigation of this relationship can be viewed as the spectral analysis of arithmetic groups. The ultimate goal of this analysis is to show that the groups in question are commensurable, i.e. share a common finite index subgroup. This has been accomplished in a number of cases. The geometric consequence of this work is the conclusion that in those cases isospectral (and more generally, length-commensurable) locally symmetric spaces are commensurable. Prior to the work of G.Prasad and the PI results of this kind were available only for 2- and 3-dimensional arithmetically defined hyperbolic manifolds. Informally, this means that the results of the project have identified many other situations where one can indeed hear the shape, at least in a weak form. The project also contributes to investigation of the structure of arithmetic and more general linear groups in the form of the congruence subgroup problem. Here one aspires to show that the normal subgroups of the group under consideration come in a natural way from the ideals of the coefficient ring. The project establishes a result in this direction for Chevalley groups of rank > 1 over arbitrary commutative rings. Furthermore, a result of this kind is obtained in the project for the automorphism group of the free group of rank 2, which can be viewed as a ``nonlinear'' analog of the group of integer 2x2-matrices. In summary, the results of the project deal with the spectral and structural analysis of arithmetic groups and their applications to differential geometry. The broader impact of the project can be seen in that the work of G.Prasad and the PI on weak commensurability ties together several areas of mathematics: in addition to the theory of algebraic and Lie groups, it uses algebraic and transcendental number theory, and in turn its results have applications to differential geometry. Second, the results of the project are likely to have impact on other areas, e.g. results on the congruence subgroup problem are relevant for the theory of automorphic forms. Third, the PI worked on some of the problems described in the project with his graduate students, so the project provided a variety of training opportunities.
