Normal Subgroup Structure of the Groups of Rational Points of Algebraic Groups and of Their Special Subgroups

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

This award provides funding for an investigation of the normal subgroup structure of groups of rational points of algebraic groups over general as well as over special (primarily, global) fields. The principal investigator attempts to prove a new conjecture on solvability of finite quotients of the groups of rational points of simple algebraic groups over arbitrary infinite fields. He also plans to analyze finite quotients of the multiplicative group of a finite dimensional division algebra in order to obtain their reasonable classification. Another central problem is investigation of the Margulis-Platonov conjecture for special unitary groups over global fields. The principal investigator collaborates on these problems with Gopal Prasad, Yoav Segev and Gary Seitz. He also plans to work with Gopal Prasad on a joint book project in the congruence subgroup problem. Questions related to the normal subgroup structure of linear groups have historical roots in the 19th century (Galois, Jordan, Dixon), and have been an area of active research in the 20th century. Recently, new techniques for analyzing this problem for anisotropic groups over general fields have been discovered in the joint work of the principal investigator with Y.Segev and G.Seitz. These techniques enable one to investigate groups over general fields using methods of the theory of valuations, which were previously used only in the number-theoretic setting of global fields. This approach fits into the general area of investigation of the congruence subgroup problem, which is connected with other fundamental problems in number theory, currently applied in data transmission, data processing and communication systems.