Timeline for Surface groups and non separating loops
Current License: CC BY-SA 3.0
10 events
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| Jul 26, 2012 at 10:07 | comment | added | HJRW | Mark - yes, I'm aware of that. If you look at the UPDATE in my answer, you'll see that I assert that 'there may well be examples of such maps with no simple loops in the kernel'. I also suggest somewhere to look for examples. | |
| Jul 24, 2012 at 15:00 | comment | added | user6976 | @Henry: The question in fact has nothing to do with 3-manifolds. What one needs is to kill long and complicated self-intersecting loops and at the same time kill the first homology by imposing relations of the form $x=$ long commutator where $x$ is a standard generator of the surface group. The goal is to produce a quotient with no simple loops in the kernel. I think it is quite doable and the example should exist. I am not sure now about the number of generators ($\le 3$) of the quotient, but that also should be possible to achieve with some effort. | |
| Aug 23, 2011 at 15:02 | comment | added | Thom | Thanks Henry! I am convinced that the answer to my question is unknown. | |
| Aug 23, 2011 at 0:54 | vote | accept | Thom | ||
| Jul 20, 2012 at 23:43 | |||||
| Aug 22, 2011 at 19:03 | comment | added | HJRW | A quick search of MathSciNet shows that the Simple Loop Conjecture holds for Seifert-fibred 3-manifolds and, more generally, graph manifolds. As there are very complicated hyperbolic 3-manifolds with trivial H_1 and two generators, I would be surprised if those assumptions help much. | |
| Aug 22, 2011 at 16:06 | comment | added | Sam Nead | @Thom - The simple loop conjecture holds for three-manifolds of the form surface cross interval, by work of Gabai. | |
| Aug 22, 2011 at 14:36 | comment | added | Thom | Thanks! Are there cases of $3$-manifolds (with infinite $\pi_1)$) for which we know Simple Loop Conjecture holds? Would it be helpful if we assume that the group $G$ has three (or less) generators and abelianization of rank 0. | |
| Aug 22, 2011 at 14:26 | vote | accept | Thom | ||
| Aug 22, 2011 at 18:12 | |||||
| Aug 22, 2011 at 12:37 | history | edited | HJRW | CC BY-SA 3.0 |
added 392 characters in body
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| Aug 22, 2011 at 12:16 | history | answered | HJRW | CC BY-SA 3.0 |