Arithmetic groups, universal lattices and Kazhdan's property (T)

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AbstractAward: DMS-0701639Principal Investigator: Yehuda ShalomThe research aspects of the project revolve around two mainrelated themes. The first pertains to Kazhdan's property oflinear groups over arbitrary finitely generated rings. Our recentresult that under certain conditions on n and the ring R, thegroup 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 promotesfundamental questions and speculations about both the nature ofthese groups, and property (T). The second topic addresses a wellknown question of Zimmer from 1979, on the subgroup structure ofarithmetic groups, which has been touched so far by only oneauthor. The question seeks refinement of Margulis' celebratednormal subgroup theorem, to the understanding of almost normalsubgroups, i.e., ones which intersect their conjugates in afinite index subgroup. We plan to pursue a new approach which onone hand leads to entirely new results and phenomena even for thesimplest arithmetic groups, and on the other to many intriguingnatural problems. The group theoretic notion of boundedgeneration turns out be deeply related to both aspects of theproject, although it arises from different sources.The mathematical notion of a group is one of the most fundamentalones in the sciences in general and in mathematics in particular,enabling one to rigorously treat as well as capitalize onsymmetries of various structures. A distinguished class ofgroups in mathematics is formed by the arithmetic groups, whichappear and are fundamental in many mathematical areas includinggeometry, number theory, algebra and even combinatorics andcomputer science. One main purpose of the project is to exposeand study entirely new features of these groups, which were notseen previously in other groups as well. Another main goal is tomake first systematic study of a wider family of groups,generalizing the arithmetic ones, and try to bring to bear someof the rich theory and powerful tools which have been developedover the past half a century in the more classical setting. Thiswould have many potential applications in those areas wherearithmetic groups have already proved to be a fundamentallyimportant object and tool.