Kazhdan's property (T) is a strong negation of amenability, relevant in the study of algebraic properties of groups, in the construction of expander graphs, in dynamics and in connection to the Baum-Connes conjectures. Various strengthened versions of property (T) have been formulated in recent years.
Among them, those involving actions on Lp spaces are particularly interesting, because of their presumed connection to the conformal dimension of the boundary of hyperbolic groups, because they manage to achieve a separation between rank one and higher rank lattices, and because of the increasing role that they play in operator algebras. In this talk I shall explain how random groups, both in the triangular model and in the Gromov model, eventually have all the strengthened Lp-versions of property (T). This is joint work
with J. Mackay.