Emmanuel Breuillard

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A very interesting large class of groups is provided by the linear groups, namely the subgroups of GL d (K), for some (commutative) field K. There are few general tools to study arbitrary finitely generated groups (often one has to resort to combinatorics and analysis as we did in Lecture 1 above for example). However for linear groups the situation is very(More)
1. Amenability, paradoxical decompositions and Tarski numbers In this exercise, we prove yet another characterization of amenability, which is due to Tarski [7, 4] and states that a group is non-amenable if and only if it is paradoxical. Let Γ be a group acting on a set X. This Γ-action is said to be N-paradoxical if one can partition X into n + m N(More)
Up until the Bourgain-Gamburd 2005 breakthrough the only known ways to turn SL d (F p) into an expander graph (i.e. to find a generating set of small size whose associated Cayley graph has a good spectral gap) was either through property (T) (as in the Margulis construction) when d 3 or through the Selberg property (and the dictionary between combinatorial(More)
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