Superrigidity for dense subgroups of Lie groups and their actions on homogeneous spaces
| Autor: | Drimbe, Daniel, Vaes, Stefaan |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Mathematische Annalen 386 (2023), 2015-2059 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s00208-022-02437-1 |
| Popis: | An essentially free group action of $\Gamma$ on $(X,\mu)$ is called W*-superrigid if the crossed product von Neumann algebra $L^\infty(X) \rtimes \Gamma$ completely remembers the group $\Gamma$ and its action on $(X,\mu)$. We prove W*-superrigidity for a class of infinite measure preserving actions, in particular for natural dense subgroups of isometries of the hyperbolic plane. The main tool is a new cocycle superrigidity theorem for dense subgroups of Lie groups acting by translation. We also provide numerous countable type $II_1$ equivalence relations that cannot be implemented by an essentially free action of a group, both of geometric nature and through a wreath product construction. Comment: v2: The examples in propositions 4.1 and 4.2 have been corrected, taking into account the exceptional isomorphisms for SO(n,m) when n+m=4. There were no other changes |
| Databáze: | arXiv |
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