HARMONICITY AND RIGIDITY OF DISCRETE AND ARITHMETIC GROUPS

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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 thisconjecture, 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.