Timeline for Finding hyperbolic metrics by approximation
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Sep 26, 2012 at 14:47 | history | edited | HenrikRüping | CC BY-SA 3.0 |
link was not working
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| Sep 5, 2012 at 12:51 | history | edited | HJRW | CC BY-SA 3.0 |
Added reference to Luo--Tillmann--Yang.
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| Apr 16, 2012 at 17:34 | comment | added | HJRW | Sam - I agree. Certainly, 3-manifoldness has little to do with 'cubulated implies virtually special'. | |
| Apr 16, 2012 at 17:18 | comment | added | Sam Nead | What I am saying is: asking for all those surface groups is a big thing to ask. Saying you can do it without analysis is, well, interesting... | |
| Apr 16, 2012 at 17:16 | comment | added | Sam Nead | Henry - I think Ian is saying that "you don't really need three-manifoldness for the statement 'cubulated implies virtually special'". So, if somebody wanders by and hands you enough surface subgroups of your hyperbolic group with sphere boundary, then you can prove that the group is virtually special. The walls will have to be surface groups again; we can use their intersection patterns to define points of the ball. The group acts on the set of intersection patterns; hence it acts on the points of the ball... | |
| Apr 16, 2012 at 17:06 | comment | added | Sam Nead | ...relying on such a nice presentation is putting the cart before the horse. To know that such a nice presentation even exists one currently has to use the Kahn-Markovic machinery - that in turn relies on knowing delicate information about the ergodicity of the geodesic flow on the unit tangent bundle to $M$, as equipped with the hyperbolic metric. So it is simply backwards to search for the nice presentation and then use it to find the nice representation. At least, that is my current understanding. | |
| Apr 16, 2012 at 16:48 | comment | added | Sam Nead | Restating my comment more directly: If I give you $G$, written as a presentation of finite index supergroup of a surface bundle group, then there at least two reasons to hope one can find the hyperbolic structure algorithmically. For example, see the work of Helling and Menzel. Also, Thurston's proof that bundles (with pA monodromy) are hyperbolic is an iterative proof -- this is called the "double limit theorem". However... | |
| Apr 16, 2012 at 16:47 | comment | added | HJRW | (Sam's speculations have now vanished, sadly.) | |
| Apr 16, 2012 at 16:46 | comment | added | HJRW | Sam - your speculations remind me of a recent, rather cryptic, comment that Ian made on my blog post about his work, viz: ldtopology.wordpress.com/2012/03/12/or-agols-theorem/… . | |
| Apr 16, 2012 at 16:26 | comment | added | HJRW | Mark - that's what I meant when I said 'it depends what you mean by 'nice'. But I don't see what a virtually-surface-bundle presentation has to do with the question. | |
| Apr 16, 2012 at 16:10 | comment | added | user6976 | @HW: Didn't Ian Agol prove that these groups are all virtually surface-by-cyclic. It can't get much nicer than that. | |
| Apr 16, 2012 at 9:43 | history | answered | HJRW | CC BY-SA 3.0 |