Principal Investigator: Yehuda Shalom
The research aspects of the project revolve around two main related themes. The first pertains to Kazhdan's property of linear groups over arbitrary finitely generated rings. Our recent result that under certain conditions on n and the ring R, the group 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 promotes fundamental questions and speculations about both the nature of these groups, and property (T). The second topic addresses a well known question of Zimmer from 1979, on the subgroup structure of arithmetic groups, which has been touched so far by only one author. The question seeks refinement of Margulis' celebrated normal subgroup theorem, to the understanding of almost normal subgroups, i.e., ones which intersect their conjugates in a finite index subgroup. We plan to pursue a new approach which on one hand leads to entirely new results and phenomena even for the simplest arithmetic groups, and on the other to many intriguing natural problems. The group theoretic notion of bounded generation turns out be deeply related to both aspects of the project, although it arises from different sources.
The mathematical notion of a group is one of the most fundamental ones in the sciences in general and in mathematics in particular, enabling one to rigorously treat as well as capitalize on symmetries of various structures. A distinguished class of groups in mathematics is formed by the arithmetic groups, which appear and are fundamental in many mathematical areas including geometry, number theory, algebra and even combinatorics and computer science. One main purpose of the project is to expose and study entirely new features of these groups, which were not seen previously in other groups as well. Another main goal is to make first systematic study of a wider family of groups, generalizing the arithmetic ones, and try to bring to bear some of the rich theory and powerful tools which have been developed over the past half a century in the more classical setting. This would have many potential applications in those areas where arithmetic groups have already proved to be a fundamentally important object and tool.
