[METRIC 2011] Emmanuel Breuillard

METRIC 2011 Trimester at Institut Henri Poincaré (Paris, France)
Workshop on Expanders and derandomization (March 21-25, 2011)
Mar 23, 13:30-14:30 - Emmanuel Breuillard (U. Paris-Sud, Orsay)
Random pairs of elements in finite simple groups of bounded rank generate expanders
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In joint work with B. Green, B. Guralnick and T.Tao, we prove that almost all pairs of elements in a finite simple group of bounded rank give rise to a Cayley graph with a uniform spectral gap. The method follows the strategy put forward a few years ago by Bourgain and Gamburd (who proved it for PSL(2,Z/pZ)) and makes use of the recent results (Pyber-Szabo, Breuillard-Green-Tao) on the classification of approximate subgroups of finite simple groups of Lie type. A key step in the proof is to show the existence of free subgroups of semisimple algebraic groups that are "strongly dense". This means that every non abelian subgroup of it is Zariski dense.