Timeline for Very particular kind of 4-manifolds. Classification
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18 events
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| Nov 6, 2020 at 20:51 | comment | added | HJRW | One final comment! Martin Bridson explained to me that the isomorphism problem for fundamental groups of aspherical 2-complexes is undecidable: this is in Chuck Miller’s thesis. Combining this fact with the construction of this answer places severe restrictions on any possible classification. | |
| Oct 10, 2020 at 3:52 | history | edited | Ian Agol | CC BY-SA 4.0 |
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| Oct 9, 2020 at 12:06 | comment | added | HJRW | Maybe I should also point out that, in the comments on an answer to the question I linked to above, Rylee Lyman gives the following example of a 2-generator, 2-relator acyclic group: $\langle a,b\mid a(aba^{-1}b^{-1})^5, b^6a^2b^3a^{-1}b^{-8}a^{-1} \rangle$. | |
| Oct 9, 2020 at 12:03 | comment | added | HJRW | @DannyRuberman, I think the canonical easiest example of an acyclic group is Higman's group. See, for example, the answers to this MO question. mathoverflow.net/questions/352015/… (Like Ian, I tend to reach for small-cancellation first!) | |
| Oct 9, 2020 at 6:20 | comment | added | Ian Agol | @DannyRuberman Hmmm, a random 2-generator, 2-relator group ought to be C'(1/6). But I think you might need fairly long relators before this kicks in. And to get a perfect group you need some condition on the degrees of the two words giving a matrix with determinant 1. In the 2-generator case, I suspect this isn't an issue since the Z^2 random walk is recurrent, although I haven't done the calculation (to make sure that it is still C'(1/6) conditioning on this). Anyway the lengths of the relators will have to be relatively long I think. | |
| Oct 8, 2020 at 18:00 | comment | added | Danny Ruberman | Is there a relatively simple presentation for the type of group you mention in your answer? If so then you could in principle draw a surgery diagram and see if Snappy or some other tool identifies the boundary. | |
| Oct 8, 2020 at 16:39 | comment | added | Denis T | I'm pretty sure that somewhere in A. J. Berrick works it's proved that there are no finitely presented acyclic groups with cd < 4, but that could be false memories. | |
| Oct 8, 2020 at 14:37 | comment | added | Ian Agol | @HJRW Okay. I guess I could point out that it doesn’t follow from Adian-Rabin since acyclicity is not preserved by taking subgroups. | |
| Oct 8, 2020 at 12:57 | comment | added | HJRW | I don't know a good reference. This is very close to the Andrews--Curtis conjecture, so one could look for references about that. One important result is the Collins--Miller's theorem that it's impossible to algorithmically decide if a 2-complex is aspherical, from their paper "The word problem in groups of cohomological dimension 2". | |
| Oct 8, 2020 at 11:56 | comment | added | Ian Agol | @HJRW Yes, I forgot to mention that. Do you have a reference? | |
| Oct 8, 2020 at 11:56 | history | edited | Ian Agol | CC BY-SA 4.0 |
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| Oct 8, 2020 at 8:15 | comment | added | HJRW | Perhaps it's worth commenting that decision problems about acyclic, aspherical 2-complexes are wide open. For instance, it's a longstanding open problem if there's an algorithm to decide whether or not such a 2-complex is simply connected. Since any meaningful "classification" should make it possible to decide if the manifold is simply connected, your answer shows that any such classification should decide this algorothmic problem too. | |
| Oct 8, 2020 at 5:10 | history | edited | Ian Agol | CC BY-SA 4.0 |
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| Oct 8, 2020 at 5:03 | vote | accept | GSM | ||
| Oct 8, 2020 at 4:12 | comment | added | Ian Agol | That's correct, I was adding a reference when you added your comment. | |
| Oct 8, 2020 at 4:11 | comment | added | Andy Putman | Isn't it not even known whether PD(3) groups are fundamental groups of 3-manifolds? I thought this was only known for PD(1) groups (Stallings) and PD(2) groups (Linnell, Muller, and Eckmann). | |
| Oct 8, 2020 at 4:11 | history | edited | Ian Agol | CC BY-SA 4.0 |
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| Oct 8, 2020 at 4:02 | history | answered | Ian Agol | CC BY-SA 4.0 |