Abstract:

The talk will be based on joint work with B. Green and T. Tao. A few years ago the initial work of H. Helfgott, who introduced such combinatorial ideas as the "sum-product phenomenon" in the study of growth in finite simple groups, was spectacularly applied by Bourgain and Gamburd to show that the Cayley graphs of the family of groups SL(2,Z/pZ), p prime, could be turned into a family of expanders for a wide range of possible choices of generating sets. These striking developments motivated the need for a more systematic study of "approximate groups", that is finite subsets of an ambient group that are almost closed under multiplication in an appropriate sense. In the talk I will present this notion due to T. Tao in some detail and survey some recent progress by a variety of authors towards the rough classification of approximate subgroups of various non-commutative groups. I will also discuss resulting applications for expanding properties of finite quotients of linear groups.