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Answering a question by Chatterji--Dru\c{t}u--Haglund, we prove that, forevery 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 ($0<p<\infty$) with unbounded orbits if and only if $p \geq p_G$. Asimilar result holds for the existence of proper continuous affine isometricactions on $L_p$ spaces. Using a representation of cohomology by harmoniccocycles, we also show that such unbounded orbits cannot occure when the linearpart comes from a measure preserving action, or more generally astate-preserving action on a von Neumann algebra and $p>2$. We also prove thestability of this critical constant $p_G$ under $L_p$ measure equivalence,answering a question of Fisher. We use this to show that for every connectedsemisimple Lie group $G$ and for every lattice $\Gamma < G$, we have$p_\Gamma=p_G$.

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