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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)

- Emmanuel Breuillard, I Expander
- 2012

There are many excellent existing texts for the material in this lecture, starting with Lubotzky's monograph [11] and recent AMS survey paper [10]. For expander graphs and their use in theoretical computer science, check the survey by Hoory, Linial and Wigderson [6]. We give here a brief introduction. We start with a definition. Definition 0.1. (Expander… (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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