2001.02490Jan 8, 2020v1: 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

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

Amine Marrakchi, Mikael de la Salle

Group Theory Functional Analysis

arXiv PDF

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 $L_p$

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space (

0<p<\infty

) with unbounded orbits if and only if

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

Γ< G

, we have

p_Γ=p_G

.

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