Timeline for Infinite group for which it is still unknown if it is simple
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 15, 2021 at 12:43 | comment | added | Uri Bader | @HJRW I believe I knew it at some point, but it has been a while since I spent my time calculating stuff about this group. I'd be happy to see you (and others) spending some time on it too, Henry ;) In any case, possibly it is described in Ronan's book cited above, but currently I don't have an access. Also, Pierre-Emmanuel Caprace or Stefan Witzel might know. | |
| Apr 15, 2021 at 12:24 | comment | added | HJRW | Thanks, Uri! Do you know what the vertex stabiliser is? (I don’t have a copy of Ronan’s book to hand...) | |
| Apr 15, 2021 at 12:17 | comment | added | Uri Bader | YCor, thanks for editing. @HJRW, as YCor said, $\Gamma$ is clearly generated by $s$ and $t$. Now, one easily observes that $s\mapsto s^2$, $t\mapsto t^2$ extends to an automorphism of $\Gamma$ which is of order 3. Constructing the corresponding semi-direct product, we get a supgroup in which $\Gamma$ is of index 3. This supgroup is isomorphic to the group $G_4$ in Ronan's "Lectures on buildings", Theorem 2.5, and $G_4$ is known to act freely transitively on the chambers of the corresponding building. I suppose this should give you a precise presentation as a triangle of groups. | |
| Apr 15, 2021 at 11:45 | comment | added | HJRW | @YCor: you're right, sorry. My question is really "what's the structure of this as a triangle of groups"? | |
| Apr 15, 2021 at 11:29 | comment | added | YCor | @HJRW of course $u$ is redundant so there's an obvious presentation on generators $s,t$. Technically this satisfies your requirement "each relator uses at most two generators", but at the same time I don't think this is what you're asking. | |
| Apr 15, 2021 at 10:49 | comment | added | HJRW | Is there by any chance a slightly more symmetrical presentation of $\Gamma$, where each relator uses at most two generators? To put it another way, I guess this is a triangle of groups... | |
| Apr 15, 2021 at 9:00 | history | edited | YCor | CC BY-SA 4.0 |
fixed typo/formatting
|
| Apr 15, 2021 at 8:36 | history | answered | Uri Bader | CC BY-SA 4.0 |