Isometric actions on Lp-spaces: dependence on the value of p

Answering a question by Chatterji--Dru\c{t}u--Haglund, we prove that, for every locally compact group $G$, there exists a critical constant $p_G \in [0,\infty]$ such that $G$ admits a continuous affine isometric action on an $L_p$ space ($02$. We also prove the stability of this critical constant $p_G$ under $L_p$ measure equivalence, answering a question of Fisher. We use this to show that for every connected semisimple Lie group $G$ and for every lattice $\Gamma < G$, we have $p_\Gamma=p_G$.
Comment: v1: 10 pages v2: 20 pages. Added : study of critical parameters for semisimple Lie groups and their lattices, and of their behaviour under quantitative measure equivalence; integrability properties of lattices v3: 24 pages. Added : discussion of harmonic cocycles and and actions on non-commutative Lp spaces coming from state-preserving actions