Timeline for Filtered colim of F-groups
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Dec 23, 2020 at 12:19 | history | bounty ended | Ofra | ||
| S Dec 23, 2020 at 12:19 | history | notice removed | Ofra | ||
| Dec 23, 2020 at 12:19 | vote | accept | Ofra | ||
| Dec 19, 2020 at 12:32 | answer | added | Matt Zaremsky | timeline score: 2 | |
| S Dec 15, 2020 at 15:32 | history | bounty started | Ofra | ||
| S Dec 15, 2020 at 15:32 | history | notice added | Ofra | Canonical answer required | |
| Dec 8, 2020 at 18:58 | comment | added | Matt Zaremsky | Let me say the second comment better. I claim Thompson's group $F$ (which is torsion-free) is not a filtered colimit of F-groups. Say it is the filtered colimit of some $(G_i)_i$ for the $G_i$ all F-groups. Then since $F$ is finitely presented it must be isomorphic to a subgroup of one of the $G_i$ (this is assuming what you're calling "filtered colimit" is the same as what I usually see called "direct limit", which I think is right, but I guess I'm not positive). But $F$ contains $\mathbb{Z}^\infty$ and so is not isomorphic to a subgroup of any F-group, a contradiction. | |
| Dec 8, 2020 at 16:24 | comment | added | Ofra | @MattZaremsky I don't really understand your second comment. | |
| Dec 8, 2020 at 16:23 | comment | added | Ofra | @MoisheKohan no, I did not. | |
| Dec 7, 2020 at 17:24 | comment | added | Matt Zaremsky | For the simpler question, I think any finitely presented torsion-free group that is not itself an F-group should be a counterexample (so like, Thompson's group $F$). Since it's finitely presented it can't be a filtered colimit in an "interesting" way (errr right? Is that how filtered colimits work?), so would have to just be an F-group itself already, which it's not. | |
| Dec 7, 2020 at 17:05 | comment | added | Moishe Kohan | For the simpler question, did you try the countably infinite direct product of ${\mathbb Z}$'s? | |
| Dec 7, 2020 at 14:51 | history | edited | Ofra | CC BY-SA 4.0 |
added 100 characters in body
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| Dec 6, 2020 at 12:05 | comment | added | Matt Zaremsky | I'm not sure what a full characterization would look like, but this feels like a pretty broad class of groups. For example it includes every torsion-free nilpotent group, every torsion-free lacunary hyperbolic group, and of course all F-groups (of which there are already a lot). Just pointing out those examples. | |
| Dec 5, 2020 at 15:12 | history | asked | Ofra | CC BY-SA 4.0 |