Timeline for What is the minimal genus of a surface acted on by the symmetric group $S_n$?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Jul 30 at 9:48 | vote | accept | André Henriques | ||
| Jul 30 at 9:48 | history | edited | André Henriques | CC BY-SA 4.0 |
added summary of relevant comments by David Speyer and Nick Gill.
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| Jul 23 at 20:30 | comment | added | André Henriques | @NickGill. Thank you to both Nick and David. Your comments completely answer my question. | |
| Jul 23 at 15:54 | comment | added | Nick Gill | Note that, subsequent to the quote in my comment above, Conder asserts that "a reasonably detailed summary was given in [4]" where [4] is the 1980 reference that @DavidESpeyer has already mentioned. | |
| Jul 23 at 15:51 | comment | added | Nick Gill | On p.745 of Regular maps with an alternating or symmetric group as automorphism group, Conder writes "It was shown by the author in his 1980 doctoral thesis [3] that both the alternating group $A_n$ and the symmetric group $S_n$ are smooth quotients of the extended $(2, 3, 7)$ triangle group for all $n\geq 167$". Note that the extended $(2,3,7)$ triangle group is the same as the $(2,3,7)$ Coxeter group. @AndréHenriques, does that deal with your query above? | |
| Jul 23 at 15:45 | comment | added | Nick Gill | @DavidESpeyer, I think you mean Conder rather than Condor. | |
| Jul 23 at 12:04 | comment | added | André Henriques | @DavidESpeyer Thanks David for the correction. So, while $S_n$ is not a $(2,3,7)$-group [i.e. it's not a quotient of the $(2,3,7)$-triangle group], it might still be a quotient of the $(2,3,7)$-Coxeter group. | |
| Jul 23 at 12:00 | comment | added | David E Speyer | Minor correction: The $(2,3,7)$ Coxeter group and the $(2,3,7)$ triangle group aren't the same thing. The former is $\langle p,q,r \rangle / \langle p^2=q^2 = r^2 = (pq)^2=(qr)^3=(rp)^7=1 \rangle$, the latter is the index $2$ subgroup generated by $x=pq$, $y=qr$, $z=rp$, with relations as in my previous comment. I think (but am not sure) that the $(2,3,7)$ Coxeter group is the one which Condor gives a different presentation for on page 1 of his paper. | |
| Jul 23 at 11:55 | comment | added | David E Speyer | @AndréHenriques Higman's paper is unpublished, but Condor has a published paper where he makes Higman's bound explicit: Condor, "Generators for alternating and symmetric groups" (1980) shows that $A_n$ is a Hurwitz group for $n \geq 168$. | |
| Jul 23 at 11:54 | comment | added | David E Speyer | We can't get equality in the Hurwitz bound. Equality happens for a group $G$ iff $G$ is generated by elements $x$, $y$, $z$ obeying $x^2=y^3=z^7=xyz=1$. But $y^3=z^7=1$ forces $y$ and $z$ in $A_n$, and then $x = z^{-1} y^{-1}$ forces $x$ in $A_n$. | |
| Jul 23 at 11:54 | comment | added | André Henriques | The question of whether the optimal bound can be attained is equivalent to the question of whether the finite group $G$ can be written as a quotient of the $(2,3,7)$ Coxeter group. I.e., whether it's a (2,3,7)-group. So my question is: is the symmetric group a (2,3,7)-group? Apparently, G. Higman has an unpublished paper in which he showed that every sufficiently large alternating group is a (2, 3, 7)-group... | |
| Jul 23 at 11:03 | history | answered | Sam Nead | CC BY-SA 4.0 |