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 $p_G \in [0,\infty]$suchthat$G$admitsacontinuousaffineisometricactiononan$L_p$space($0<p<\infty$)withunboundedorbitsifandonlyif$p \geq p_G$. A similar result holds for the existence of proper continuous affine isometric actions on $L_p$ spaces. Using a representation of cohomology by harmonic cocycles, we also show that such unbounded orbits cannot occure when the linear part comes from a measure preserving action, or more generally a state-preserving action on a von Neumann algebra and $p>2$. 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 $\mathrm{\Gamma }<G$, we have $p_{\mathrm{\Gamma }}=p_G$.