On growth of cocycles of isometric representations on Lp-spaces

We study different notions of asymptotic growth for 1-cocycles of isometric representations on Banach spaces. One can see this as a way of quantifying the absence of fixed point properties on such spaces. Inspired by the work of Lafforgue, we show the following dichotomy: for a compactly generated group G, either all 1-cocycles of G taking values in Lp[jls-end-space/]-spaces are bounded (this is Property FLp[jls-end-space/]) or there exists a 1-cocycle of G taking values in an Lp[jls-end-space/]-space with relatively fast growth. We also obtain upper and lower bounds on the average growth of harmonic 1-cocycles with values in Banach spaces with convexity properties. As a consequence, we obtain bounds on the average growth of all 1-cocycles with values in Lp[jls-end-space/]-spaces for groups with property (T). Lastly, an appendix revisits techniques by Naor and Peres in the locally compact setting, in order to show that for a compactly generated group G, the existence of a 1-cocycle with compression larger than n implies the Liouville property for a large family of probability measures on G.