Timeline for Gluing hexagons to get a locally CAT(0) space
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 21, 2019 at 12:57 | answer | added | HJRW | timeline score: 3 | |
| Nov 4, 2018 at 14:51 | history | edited | Dylan Thurston | CC BY-SA 4.0 |
added 42 characters in body
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| Nov 4, 2018 at 3:17 | history | edited | Dylan Thurston | CC BY-SA 4.0 |
added 161 characters in body
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| Nov 4, 2018 at 3:16 | comment | added | Dylan Thurston | @HJRW, indeed I missed those! There are several gluings that give that surface. | |
| Nov 3, 2018 at 7:50 | comment | added | HJRW | Wait, what about the surface of Euler characteristic -1? Shouldn’t that be on your list? | |
| Nov 2, 2018 at 19:25 | comment | added | Dylan Thurston | My motivation was very simple: I wanted an example for my class on Metric Geometry. But then I was surprised I hadn't seen these examples. | |
| Nov 2, 2018 at 6:24 | history | edited | ThiKu | CC BY-SA 4.0 |
typo
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| Nov 2, 2018 at 4:22 | comment | added | Victor Protsak | @Ian Agol: Right, and your descriptions of these groups as free-by-cyclic groups yield a lot of extra information. I am not sure what Dylan's motivations were, but I thought of algorithmically distinguishing these groups. Then it may be useful to have a simple criterion based on residual properties (or quotients). | |
| Nov 1, 2018 at 14:10 | comment | added | Ian Agol | @VictorProtsak my answer shows that the kernels of the maps to $\mathbb{Z}$ have distinct ranks, so the groups are different. | |
| Nov 1, 2018 at 13:29 | vote | accept | Dylan Thurston | ||
| Nov 1, 2018 at 6:08 | comment | added | Victor Protsak | If you look at the standard single-relator presentations of the last two groups, the total powers of $a,b$ in the relator are $(1,1)$ and $(1,3) $. While these groups have the same abelianization, it seems plausible that they can be distinguished by their homomorphisms into the finite Heisenberg group over $\mathbb{F}_3$.This should be easily checkable by a computer algebra system. | |
| Nov 1, 2018 at 5:46 | answer | added | Ian Agol | timeline score: 17 | |
| Nov 1, 2018 at 4:44 | comment | added | Ian Agol | I'm certain that one of the last two must be a spine of the Gieseking manifold. See: arxiv.org/abs/1008.1468 In any case, they should be 1-relator small-cancellation groups, so ought to be known at least to group theorists. | |
| Nov 1, 2018 at 2:25 | comment | added | Dylan Thurston | To follow up on the "not a surface" comment, the link of the vertex is a tetrahedron, with edge lengths of 2pi/3. Thus the shortest cycle is of length 2pi, so the link is CAT(1) and the space is locally CAT(0). | |
| Nov 1, 2018 at 1:52 | comment | added | Lee Mosher | You just glue them. It's not a surface. | |
| Oct 31, 2018 at 23:12 | comment | added | Steven Stadnicki | Maybe I'm missing something obvious here, but how do you glue three edges together? | |
| Oct 31, 2018 at 19:41 | history | asked | Dylan Thurston | CC BY-SA 4.0 |