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# Documents  de la Salle, Mikael | enregistrements trouvés : 2

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## $L^2$ spectral gap and group actions on Banach spaces de la Salle, Mikael | CIRM H

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Research talks;Analysis and its Applications;Geometry

Exploring the relations between algebraic and geometric properties of a group and the geometry of the Banach spaces on which it can act is a fascinating program, still widely mysterious, and which is tightly connected to coarse embeddability of graphs into Banach spaces. I will present a recent contribution, joint with Tim de Laat, where we give a spectral (hilbertian) criterion for fixed point properties on uniformly curved Banach spaces.

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## Group actions on Lp spaces : dependence on p de la Salle, Mikael | CIRM H

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Research talks;Analysis and its Applications

The study of group actions on Hilbert spaces is central in operator algebras, geometric group theory and representation theory. In many natural situations however, particularily interesting actions on Lp spaces appear for p not 2. One celebrated example is the construction by Pansu (and later greatly generalized by Yu to all Gromov hyperbolic groups) of proper actions of groups of isometries of hyperbolic spaces on Lp for large p. In all these results, the rather clear impression was that it was easier to act on Lp space as p becomes larger. The goal of my talk will be to explain this impression by a theorem and to study how the behaviour of the group actions on Lp spaces depends on p and on the group. In particular, I will show that the set of values of p such that a given countable groups has an isometric action on Lp with unbounded orbits is of the form $[p_c,\infty]$ for some $p_c$, and I will try to compute this critical parameter for lattices in semisimple groups. In passing, we will have to discuss how these objects and properties behave with respect to quantitative measure equivalence. This is a joint work with Amine Marrakchi, partly in arXiv:2001.02490. The study of group actions on Hilbert spaces is central in operator algebras, geometric group theory and representation theory. In many natural situations however, particularily interesting actions on Lp spaces appear for p not 2. One celebrated example is the construction by Pansu (and later greatly generalized by Yu to all Gromov hyperbolic groups) of proper actions of groups of isometries of hyperbolic spaces on Lp for large p. In all these ...

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