Timeline for Properties (T) and (FA)
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| May 21, 2021 at 15:57 | history | edited | AGenevois | CC BY-SA 4.0 |
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| May 21, 2021 at 9:19 | comment | added | AGenevois | @HJRW: Sure, thank you. | |
| May 21, 2021 at 9:19 | history | edited | AGenevois | CC BY-SA 4.0 |
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| May 21, 2021 at 9:09 | comment | added | HJRW | I think the implication arrows in the final line should be the other way around... | |
| May 21, 2021 at 9:07 | comment | added | AGenevois | @Aurel: Finitely generated torsion groups have $(FW_\omega)$ but they may not have $(FW)$, e.g. some infinite free Burnside groups. Thompson's groups $T$ and $V$ have $(FW_\infty)$ but they do not have $(FW_\omega)$. | |
| May 21, 2021 at 8:44 | comment | added | Aurel | @YCor Thanks, that's already a nice separation! | |
| May 21, 2021 at 8:12 | comment | added | YCor | @Aurel: $SL_2(\mathbf{Z}[\sqrt{2}]$ is FW but not T. The group $(\mathbf{Z}^n)_0\rtimes Sym(n)$ (index zero means one looks at $n$-tuples of sum zero) has FW$_{k(n)-1}$ but not FW$_{k(n)}$ for some $k(n)$, and certainly $k(n)$ tends to infinity (I'm not sure of its value), so certainly this infinite family is "essentially" separated, and with some effort it should be entirely separated using virtually abelian groups. The Grigorchuk group has FW$_\infty$ but not FW, I'm not sure about FW$_\omega$. | |
| May 21, 2021 at 7:46 | comment | added | Aurel | Are there known groups that separate all the properties in this hierarchy? | |
| May 21, 2021 at 6:48 | comment | added | YCor | The result mentioned is mostly due to Robertson-Steger for actions of discrete groups (the most difficult part is to pass from an action on a Hilbert to an action on $L^1$-space, which is median — CDH did it for actions of locally compact groups, for which there was a technical difficulty). At that time (around 2005) the notion of "measured wall spaces" was fashionable and the characterization of Property T in these terms was already known (for discrete groups, and explicitly open for locally compact groups). | |
| May 21, 2021 at 6:44 | comment | added | YCor | I've used in this paper the terminology "Property FM" for a property strictly weaker than Property T (every action on a discrete set preserving a mean has a finite orbit). | |
| May 21, 2021 at 5:33 | history | answered | AGenevois | CC BY-SA 4.0 |