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Amenable actions and rigidity.


Given a homomorphism F of a finitely generated group G into the group SL(n,Z) with Zariski dense image one can ask for counting elements of G whose images under F have some prescribed properties. This question is well understood if counting in G means counting generic elements with respect to a random walk on G. We explain the underlying principle and give a different and more general perspective in terms of amenable group actions. The main application consists in counting results for the mapping class group of a closed surface.