No CrossRef data available.
Article contents
Isometric actions on Lp-spaces: dependence on the value of p
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
- 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
References
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230811000808026-0182:S0010437X23007121:S0010437X23007121_inline899.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230811000808026-0182:S0010437X23007121:S0010437X23007121_inline900.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230811000808026-0182:S0010437X23007121:S0010437X23007121_inline902.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230811000808026-0182:S0010437X23007121:S0010437X23007121_inline903.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230811000808026-0182:S0010437X23007121:S0010437X23007121_inline904.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230811000808026-0182:S0010437X23007121:S0010437X23007121_inline905.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230811000808026-0182:S0010437X23007121:S0010437X23007121_inline907.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230811000808026-0182:S0010437X23007121:S0010437X23007121_inline911.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230811000808026-0182:S0010437X23007121:S0010437X23007121_inline912.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230811000808026-0182:S0010437X23007121:S0010437X23007121_inline913.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230811000808026-0182:S0010437X23007121:S0010437X23007121_inline915.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230811000808026-0182:S0010437X23007121:S0010437X23007121_inline916.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230811000808026-0182:S0010437X23007121:S0010437X23007121_inline917.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230811000808026-0182:S0010437X23007121:S0010437X23007121_inline918.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230811000808026-0182:S0010437X23007121:S0010437X23007121_inline920.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230811000808026-0182:S0010437X23007121:S0010437X23007121_inline921.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230811000808026-0182:S0010437X23007121:S0010437X23007121_inline922.png?pub-status=live)