Timeline for A non-hyperfinite type III factor from an action of the free group on the circle
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Aug 31, 2018 at 17:00 | comment | added | Sebastien Palcoux | @JessePeterson: I just edited a proof of the non-amenability of the action, using the last paper of Laurent Bartholdi. I also proved that it is a ${\rm III}_1$ factor in answer. | |
| Aug 31, 2018 at 16:50 | vote | accept | Sebastien Palcoux | ||
| Aug 30, 2018 at 20:34 | answer | added | Sebastien Palcoux | timeline score: 2 | |
| Aug 28, 2018 at 12:39 | comment | added | Sebastien Palcoux | @მამუკაჯიბლაძე: Right. It is important to understand the word "essentially" as "up to a subset of null measure". Recall that an action is essentially free if $\lambda$-almost every point has a trivial stabilizer, namely $\lambda(\{ x \in X \ | \ G_x \neq 1 \}) = 0$. | |
| Aug 28, 2018 at 11:09 | comment | added | მამუკა ჯიბლაძე | I find the terminology "essentially free" confusing: it suggests that the action must be close in some sense to a free action, i. e. to an action with trivial stablizers. But the definition is such that it does not in principle exclude the case when there is a point stabilized by the whole group. | |
| Aug 28, 2018 at 10:12 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
A proof of the non-amenability of the action, using the last paper of Laurent Bartholdi.
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| Sep 12, 2013 at 14:22 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
I write about the incompleteness of argument (d)
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| Sep 8, 2013 at 17:05 | comment | added | André Henriques | ... is equipped with an action of $\mathbb R_{>0}$ (coming from the action on $\Omega^{top}_{>0}(M)$). This corresponds to a action of $\mathbb R_{>0}$ on some measure space $X$. If that action is transitive, it is equivalent to $\mathbb R_{>0}$ acting on $\mathbb R_{>0}/\mathbb Z^\lambda$ for some $\lambda\in(0,1)$, and the factor $L^\infty(M)\rtimes\Gamma$ is of type $III_\lambda$. Otherwise, $L^\infty(M)\rtimes\Gamma$ is of type $III_0$. | |
| Sep 8, 2013 at 16:58 | comment | added | André Henriques | I don't know the literature, so I can't point to a reference. But here's how things go: given an (let's say a.e. smooth) action of a group $\Gamma$ on a manifold $M$, you can form the bundle of densities $\Omega^{top}_{>0}M$, which is a principal bundle with structure group $\mathbb R_{>0}$. The action of $\Gamma$ on $M$ induces an action on $\Omega^{top}_{>0}M$, and the vN algebra $L^\infty(M)\rtimes \Gamma$ is a type $III_1$ factor iff the action of $\Gamma$ on $\Omega^{top}_{>0}M$ is ergodic. If that action is not ergodic, the vN algebra $L^\infty(\Omega^{top}_{\>0}M)^\Gamma$... | |
| Sep 8, 2013 at 15:09 | comment | added | André Henriques | I'm pretty sure that it's type $III_1$, because the following set is dense in $\mathbb R_{>0}$: $\{f'(x) : x\in \mathbb S^1, f\in F_2, f(x)=x\}$. | |
| Sep 8, 2013 at 9:31 | history | asked | Sebastien Palcoux | CC BY-SA 3.0 |