Measurable splittings and the measured group theoretic structure of wreath products
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912128376504320 |
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| author | Tucker-Drob, Robin Wróbel, Konrad |
| author_facet | Tucker-Drob, Robin Wróbel, Konrad |
| contents | Let $Γ$ be a countable group that admits an essential measurable splitting (for instance, any group measure equivalent to a free product of nontrivial groups).
We show: (1) for any two nontrivial countable groups $B$ and $C$ that are measure equivalent, the wreath product groups $B\wrΓ$ and $C\wrΓ$ are measure equivalent (in fact, orbit equivalent) -- this is interesting even in the case when the groups $B$ and $C$ are finite; and (2) the groups $B\wr Γ$ and $(B\times\mathbf{Z})\wrΓ$ are measure equivalent (in fact, orbit equivalent) for every nontrivial countable group $B$.
On the other hand, we show that certain wreath product actions are not even stably orbit equivalent if $Γ$ is instead assumed to be a sofic icc group that is Bernoulli superrigid, and $B$ and $C$ have different cardinalities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11754 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Measurable splittings and the measured group theoretic structure of wreath products Tucker-Drob, Robin Wróbel, Konrad Group Theory Dynamical Systems Logic Operator Algebras 37A20 (Primary) 20E22, 28D15 (Secondary) Let $Γ$ be a countable group that admits an essential measurable splitting (for instance, any group measure equivalent to a free product of nontrivial groups). We show: (1) for any two nontrivial countable groups $B$ and $C$ that are measure equivalent, the wreath product groups $B\wrΓ$ and $C\wrΓ$ are measure equivalent (in fact, orbit equivalent) -- this is interesting even in the case when the groups $B$ and $C$ are finite; and (2) the groups $B\wr Γ$ and $(B\times\mathbf{Z})\wrΓ$ are measure equivalent (in fact, orbit equivalent) for every nontrivial countable group $B$. On the other hand, we show that certain wreath product actions are not even stably orbit equivalent if $Γ$ is instead assumed to be a sofic icc group that is Bernoulli superrigid, and $B$ and $C$ have different cardinalities. |
| title | Measurable splittings and the measured group theoretic structure of wreath products |
| topic | Group Theory Dynamical Systems Logic Operator Algebras 37A20 (Primary) 20E22, 28D15 (Secondary) |
| url | https://arxiv.org/abs/2410.11754 |