Timeline for Finite sheeted covering of the complement of a finite set in $\mathbb{C}$
Current License: CC BY-SA 4.0
13 events
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| Nov 9, 2022 at 13:02 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Oct 10, 2022 at 11:18 | answer | added | Vladimir Dotsenko | timeline score: 1 | |
| Oct 10, 2022 at 11:07 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jun 12, 2022 at 10:05 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| May 13, 2022 at 15:37 | comment | added | piper1967 | Well can I say atleast that such covering space is $\mathbb{C}$ with finitely many points removed? | |
| May 13, 2022 at 9:46 | review | Close votes | |||
| May 18, 2022 at 3:02 | |||||
| May 13, 2022 at 9:20 | answer | added | Sam Nead | timeline score: 1 | |
| May 13, 2022 at 8:45 | comment | added | YCor | @A.S. I'm not sure this is a representation-theoretic fact, but "classifying" finite index subgroups of $F_2$ is not easier than "classifying" finite index subgroups of $F_d$ for larger $d$, since $F_d$ itself lies as a finite index subgroup of $F_2$. — Note however that the OP also fixes the index ($n$ in OP's notations, assuming we stick to nonempty connected coverings), and hence one could also fix the index and let the number of generators grow. | |
| May 13, 2022 at 8:41 | comment | added | YCor | @A.S. your $n$ is not the OP's $n$ (it is $|S|$ in the OP's notation). | |
| May 12, 2022 at 21:50 | comment | added | user164898 | Isomorphism classes of finite-to-one covers of $\mathbb{C}$-with-$n$-punctures are in bijection with isomorphism classes of finite $F_n$-sets, where $F_n$ is the free group on $n$ letters. I imagine that, as $n$ grows (perhaps already for $n=2$ or $n=3$), the problem of classifying finite $F_n$-sets becomes intractable, in the sense that the representation theory of $F_n$ becomes "of wild type." Perhaps somebody reading this page who knows more rep. theory than I do can speak up about what the least value of $n$ is, such that classifying the finite $F_n$-sets is a problem "of wild type." | |
| May 12, 2022 at 21:48 | history | edited | YCor | CC BY-SA 4.0 |
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| May 12, 2022 at 21:08 | comment | added | Will Sawin | We get one connected n-sheeted covering for each $|S|$-tuple of permutations in $S_n$ that together generate a group that acts transitively on $\{1,\dots, n\}$, up to simultaneous conjugation. So really there is a reasonable classification only for $|S|=1$, where only an $n$-cycle does the trick and all $n$-cycles are conjugate. | |
| May 12, 2022 at 21:04 | history | asked | piper1967 | CC BY-SA 4.0 |