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Article contents
Isometric actions on Lp-spaces: dependence on the value of p
Part of:
Special aspects of infinite or finite groups
Selfadjoint operator algebras
Linear function spaces and their duals
Lie groups
Ergodic theory
Published online by Cambridge University Press: 26 May 2023
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 
$p \geq p_G$. 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 occur 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.
Keywords
MSC classification
Primary:
22E41: Continuous cohomology
20F65: Geometric group theory
37A40: Nonsingular (and infinite-measure preserving) transformations
46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46L52: Noncommutative function spaces
- Type
- Research Article
- Information
- Copyright
- © 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence
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