Timeline for Do surface groups embed into PSL_2 over a real quadratic integer ring?
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 18, 2021 at 23:32 | vote | accept | Ian Gershon Teixeira | ||
| Dec 16, 2021 at 12:43 | history | edited | Jean Raimbault | CC BY-SA 4.0 |
added 878 characters in body
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| Dec 15, 2021 at 17:34 | comment | added | Ian Gershon Teixeira | I accept 100% of the blame for changing the question twice and any confusion that caused. I'll edit later and put in a short history of the question also change the title as suggested. Also I think the alternative approach mentioned in paragraph 2 using hyperbolic geometry and explicit generators of the triangle group is here mathoverflow.net/questions/377958/… which links to the detailed reference neilstrickland.github.io/genus2/genus2.pdf | |
| Dec 15, 2021 at 8:25 | comment | added | Jean Raimbault | @MoisheKohan : i posted a short argument for the existence of $\mathbb Q$-representations below ; i was not aware of the paper of Edmonds--Ewing--Kulkarni, thanks for the reference. | |
| Dec 14, 2021 at 22:41 | comment | added | Moishe Kohan | Incidentally, for the (2,4,6)-triangle group, the smallest genus is 6. See the references in my answer here. | |
| Dec 14, 2021 at 18:12 | comment | added | Moishe Kohan | @JeanRaimbault: Could you expand your answer by including a proof of existence of rational faithful representations of surface groups as in your original comments? I was unaware of this fact and, I think, it should be better known. | |
| Dec 14, 2021 at 17:13 | comment | added | HJRW | @MoisheKohan: Well, I agree that the exact question has got a bit lost in all this. Still, it looks like the OP recognises that they should look at Maclachlan--Reid, which perhaps is the most important outcome! | |
| Dec 14, 2021 at 16:37 | comment | added | Moishe Kohan | @HJRW: I did not mean to disrespect the answer (which I upvoted). My comment came from the fact that the question was heavily edited and is now quite a bit different from what was originally asked (I did not realize this when commenting under the current answer). The original question had a different answer, see comments underneath the question itself. | |
| Dec 14, 2021 at 16:33 | comment | added | Jean Raimbault | @MoisheKohan yes, there are now 3 versions of the question. Maybe OP (or somebody else) should preserve the history in the question. Also the title is now a bit too vague for the question (it could be "do surface groups embed into PPSL_2 over a real quadratic integer ring?") | |
| Dec 14, 2021 at 16:31 | comment | added | Moishe Kohan | @JeanRaimbault: Oh, I see, this is different from the original question, where OP asked only about an embedding in $PSL(2,F)$, $F$ a number field. | |
| Dec 14, 2021 at 16:29 | comment | added | HJRW | @MoisheKohan: Your comment "all surface groups embed over real quadratic fields, bit the proof is different from what you wrote" comes across as somewhat disrespectful. This answer gives a proof. You may prefer a different proof. Neither is "the" proof. | |
| Dec 14, 2021 at 16:26 | comment | added | Jean Raimbault | @Moishe Kohan : the point is to get an embedding into the integral points $PSL_2(\mathbb Z_F)$, this is not possible for $F=\mathbb Q$ since all subgroups of $PSL_2(\mathbb Z)$ are virtually free. | |
| Dec 14, 2021 at 13:41 | comment | added | Ian Gershon Teixeira | oh wow that comment about being able to embed some surface groups in $ PSL(\mathbb{Z}[\sqrt{2}]) $ has really piqued my curiosity. Now I'm changing the question to ask for embedding into a quadratic extension of $ \mathbb{Z} $. Obviously your answer is super awesome and I'll accept it if no one wants to talk about quadratic extensions of the integers. But I want to leave the (new edited) question open a little longer to see if I can find out more knowledge (and I upvoted you of course!) (and I think I might get that Maclachlan-Reid book now for the holidays!) | |
| Dec 14, 2021 at 13:36 | comment | added | Moishe Kohan | Yes, all surface groups embed over real quadratic fields, bit the proof is different from what you wrote. However, what's wrong with quoting Takeuchi and getting a faithful representation over Q? | |
| Dec 14, 2021 at 7:33 | history | answered | Jean Raimbault | CC BY-SA 4.0 |