Timeline for Existence of a surface group ensures the existence of a $\pi_1$-injective immersed surface
Current License: CC BY-SA 4.0
10 events
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| Apr 16 at 19:35 | comment | added | Ian Agol | If you don’t require the surface to be immersed, then there is an elementary argument giving a map of a surface which is $\pi_1$-injective. Just take a cell structure on the surface, map the i-skeleton in by induction. | |
| Apr 12 at 6:45 | vote | accept | one potato two potato | ||
| Apr 12 at 6:42 | comment | added | HJRW | @onepotatotwopotato: Yes! (At least in the hyperbolic case, given your notation.) If you have a reference for that fact, it will give the extra fact you ask for in the hyperbolic case. | |
| Apr 12 at 2:57 | comment | added | one potato two potato | In the last paragraph, you said "one can argue that $N_0$ is an interval bundle." is it because the topological type of elements in $AH(\pi_1(N_0))$ (in case of hyperbolic) is an interval bundle? | |
| Apr 11 at 12:36 | comment | added | HJRW | @SamNead: yes, this is a detail of the proof that I have swept under the rug (in my appeal to the proof of Scott's theorem). One deals with this by passing to the pieces of the Kneser--Milnor decomposition. But the upshot is that the result is true in general. | |
| Apr 11 at 11:47 | comment | added | Sam Nead | A small point - If we don't assume that $M$ is hyperbolic, then we don't know that $M$ is irreducible. So some boundary components of $N_0$ could be spheres. Or am I confused? | |
| Apr 11 at 7:25 | history | edited | HJRW | CC BY-SA 4.0 |
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| Apr 11 at 7:23 | comment | added | HJRW | This is probably written somewhere, though I don’t know a good reference. Perhaps one of the 3-manifold experts active on MO can suggest something. I had in mind examining the JSJ decomposition of $N_0$ (along the lines of this paper: arxiv.org/abs/math/9712227v2) and ruling out all the cases except the case of an interval bundle. | |
| Apr 11 at 6:29 | comment | added | one potato two potato | Thanks. After reading your answer I actually wondered about the existence of an immersed surface $S$ whose $\pi_1$ is $\Gamma$ (written your last paragraph). Could you explain in more detail how I can find such an immersed surface? | |
| Apr 11 at 6:16 | history | answered | HJRW | CC BY-SA 4.0 |