Fixed point properties for group actions on Banach spaces

Group actions on Banach spaces (in particular on Lp-spaces) have seen a growing interest in the last decades. After an introduction to this topic, I will explain a joint work with Mikael de la Salle, in which we established a spectral criterion for fixed point properties of group actions on large classes of Banach spaces (including Lp-spaces). This criterion can be applied to random groups in certain models. Our work also lead to new estimates on the conformal dimension of the boundary of random groups.