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Romain Tessera

Amenability, affine isometric actions and embedding of trees in Banach spaces

plan of the course:

1. introduction to amenable groups (Von Neumann and Folner criterion, stability properties, examples and counterexamples)

2. large-scale geometry of locally compact groups, and Cayley-Abels graphs. Example of amenable group whose CA graph is a tree: the affine group over Q_p.

3. Isometric actions on normed vector spaces. Relation between displacement of the action, and distortion of the orbits.

4. Central result: a Banachic version of a result of Delorme for isometric actions of the affine group over Q_p.

5. Corollary: a new and original proof of Bourgain's theorem that a 3-regular tree does not embed bi-lipschitz into a superreflexive Banach space.