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

Abstract

Answering a question by Chatterji--Druţu--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 ($0<p<\infty$) with unbounded orbits if and only if $p\ge p_G$. 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 $\Gamma <G$, we have $p_{\Gamma }=p_G$.

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