Arithmetic Groups, Their Applications and Generalizations

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

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.