Timeline for Conditions on the hierarchy for Thurston's hyperbolization theorem
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 12, 2015 at 8:23 | vote | accept | Don Shanil | ||
| Oct 12, 2015 at 8:23 | vote | accept | Don Shanil | ||
| Oct 12, 2015 at 8:23 | |||||
| Oct 11, 2015 at 12:08 | comment | added | Danny Ruberman | If M is closed, the hyperbolic structure is unique, so independent of hierarchy. The argument is inductive, which is where the condition on annuli comes in, to provide the correct topological hypotheses at each stage of cutting along a surface in the hierarchy. As Ian commented above, the argument is different if M fibers over the circle. | |
| Oct 11, 2015 at 12:05 | history | edited | Danny Ruberman | CC BY-SA 3.0 |
added 99 characters in body
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| Oct 11, 2015 at 6:29 | comment | added | Don Shanil | Thanks for the reference. I could be muddling things up, but isn't the conditions of no incompressible tori/annuli a condition on the manifold rather than anything to do with the hierarchy? To put it another way, suppose I have two explicitly given hierarchies $\mathcal{H}_1$ and $\mathcal{H}_2$ for a manifold $M$ that satisfies all conditions for being a hyperbolic $3$--manifold. If I cut up and re-glued each of these in turn, can it happen I end up with a hyperbolic structure on $(M,\mathcal{H}_1)$ but not on $(M,\mathcal{H}_2)$. | |
| Oct 10, 2015 at 14:21 | history | answered | Danny Ruberman | CC BY-SA 3.0 |