Timeline for Are mapping class groups of orientable surfaces good in the sense of Serre?
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| Sep 30, 2018 at 14:04 | comment | added | Ian Agol | @Tsein32 yes, I forgot in the case of one puncture, the kernel is a surface group, not free. In any case, the same finiteness result holds. More generally, it will be true for any group with a finite $K(\pi,1)$. | |
| Sep 30, 2018 at 13:20 | comment | added | Tsein32 | I figured the argument would be something like that. Does this also hold for surface groups though? If I understand correctly, you use that to get goodness of the mapping class group of $S\setminus\lbrace x\rbrace$ from Lemma 3.3 when S is closed and has genus greater than 0. | |
| Sep 30, 2018 at 13:10 | comment | added | Ian Agol | @Tsein32 $N$ in this case is a finitely generated free group, so $H^q(N,M)$ is finite for $M$ finite (it’s essentially the homology of the kernel of the action on $M$). | |
| Sep 30, 2018 at 8:12 | comment | added | Tsein32 | Thank you for this very detailed comment. I've been trying to flesh out your argument for goodness of the closed mapping class group implying goodness for the punctured mapping class group. The one thing I can't figure out is why $\pi_1(S_{g,n})$ satisfies the extra conditions on $N$ in Lemma 3.3 of the paper you linked. Is there just some general theorem I'm missing for when the cohomology groups $H^q(N,M)$ are finite for finite $M$? | |
| Sep 18, 2018 at 14:59 | history | edited | Ian Agol | CC BY-SA 4.0 |
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| Sep 18, 2018 at 5:12 | history | edited | Ian Agol | CC BY-SA 4.0 |
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| Sep 18, 2018 at 4:50 | history | answered | Ian Agol | CC BY-SA 4.0 |