### Group actions on L_p spaces : dependence on p

Analysis and Geometry Seminar

5th November 2020, 3:15 pm – 4:15 pm

Online, (contact organisers for details)

The study of group actions on Hilbert spaces is central in operator algebras, geometric group theory and representation theory and has many applications to other areas of mathematics. In many natural situations however, particularily interesting actions naturally appear on more general Banach spaces, and in particular on L_{p} spaces 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 L_{p} for large p. In all these results, the rather clear impression was that it was easier to act on L_{p} 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 L_{p} 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 L_{p} with unbounded orbits is of the form [p_{c},∞] 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.

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