hermes.lajoinie@umontpellier.fr
A group has Property (T) if every action by an affine isometry on a Hilbert space admits a fixed point. This definition emphasizes the idea that group actions having Property (T) are rigid. In his work on the Baum-Connes conjecture, Vincent Lafforgue defined in 2007 a strengthening of Property (T) that implies a fixed point result for affine actions on Hilbert spaces that are no longer isometric, but for which the operator norm growth is sub-exponential. Lafforgue also showed that any action by an isometry on a uniformly locally finite Gromov-hyperbolic space of a group having the strong Property (T) admits bounded orbits.
I will present a work where I show that relatively hyperbolic groups do not have the strong Property (T). The idea of the proof, similar to Lafforgue's proof for hyperbolic groups, is to use a natural action on a hyperbolic graph to construct a representation of our group into a Hilbert space that has sub-exponential growth and no fixed point.