Affine actions, cohomology and hyperbolicity

Many groups admit an affine action on a Hilbert, or suitable Banach space which is proper, or at least has an unbounded orbit. For example, CAT(0) cubical groups act properly on Hilbert space; and a hyperbolic group acts properly on an Lp-space, although some cannot act on a Hilbert space with an unbounded orbit (or even without a global fixed point). In the talk I will describe these results and will discuss some recent work, joint with Eric Reckwerdt and Romain Tessera, on the existence of affine actions of relatively hyperbolic groups.