The principal aim of the proposed activity is to enhance research in the rigidity theory of discrete groups, with particular emphasis on the important class of arithmetic groups. Substantial parts of the research aim to sharpen, recast and unify diverse results in the theory of arithmetic groups into one novel setting, through a concrete general conjecture, with particular emphasis on the 30-year old Margulis-Zimmer conjecture (concerning the structure of commensurated subgroups), as a special case. A key notion in the proposed work is that of harmonicity, which emerges as central ingredient in relation to this conjecture, as well as in a different suggested direction of rigidity pertaining to Lipschitz harmonic functions on groups (motivated by their use in recent joint work of the PI with Terry Tao).
The theory of so-called arithmetic groups is one of the deepest, most beautiful, fruitful, and long studied ones in modern mathematics. It involves most fields of pure mathematics, from the algebraic side of representation, number, and K-theory, through geometry in its various forms, to the analytic side of ergodic theory and dynamics. The proposed work aims to address some fundamental issues and suggest a recast of parts of the theory, thereby shedding new light and enhancing new developments in the field. Of particular emphasis is a 30-year old conjecture due to Gregory Margulis and Robert Zimmer. Its resolution is expected to involve new dynamical ideas, which are strongly related to harmonicity (roughly speaking, a state which is stable under averaging). Novel investigations and applications of harmonicity are expected to bring progress in other directions of the proposed work.
In the scientific part of the project major advances were made towards the solution of some fundamental problems in the (so-called rigidity) theory of infinite groups-- these are objects of fundamental importance all over mathematics, as well as in other areas of science including physics and chemistry. One of the problems is a fundamental conjecture due to Gregory Margulis and Robert Zimmer from the late 70's, which resisted most attempts so far (with essentially one paper attacking special cases of it, due to Venkataramana). Together with George Willis we were able to cover more cases, and present a novel approach which is able to prove much more for them. Our approach also promotes a far reaching conjecture which unifies in a surprising manner very different aspects of the theory. Thus, the problems we considered relate and intertwine different areas of mathematics ( geometry, probability algebra and number theory to name a few). Our work had contributed to the creation of new connections between those themes. Some of our findings were already published in scientific journals. The funding of this project has also been used to support graduate students, some of whom were collaborated with in the research on the above mentioned problems.
