Isometric actions on <i>L</i>_<i>p</i>-spaces: dependence on the value of <i>p</i>

Answering a question by Chatterji–Druţu–Haglund, we prove that, for every locally compact group $G$ , there exists critical constant $p_G \in [0,\infty ]$ such that admits continuous affine isometric action on an $L_p$ space ( $0&lt; p&lt;\infty$ ) with unbounded orbits if and only $p \geq p_G$ . A similar result holds the existence of proper actions spaces. Using representation cohomology harmonic cocycles, also show cannot occur when linear part comes from measure-preserving action, or more generally state-preserving von Neumann algebra $p&gt;2$ We stability this $p_G$ under measure equivalence, answering Fisher.