Final Report Summary - GADA (Group Actions: Interactions between Dynamical Systems and Arithmetic)
Our aim in this project was to apply a number of emerging tools from combinatorics and dynamical systems to certain classical problems in geometric group theory and spectral geometry. A key aspect of the developments that arose during the five year period of the project was the systematic study of approximate structures in group theory. In particular the notion of an approximate subgroup. We imported several powerful tools and ideas from that part of combinatorics related to additive number theory, to the study of arithmetic progressions, to graph theory, in the spirit of the celebrated works of Szemeredi's and Green-Tao's, and showed how to adapt them to the new context of group theory and Riemannian geometry. This lead to surprising connections with part of Gromov's work on Riemannian geometry and geometric group theory, but also with the asymptotic theory of finite groups, the study of geometrical and spectral properties of large finite simple groups, culminating with proofs of conjectures of Lubotzky on expander graphs.