Timeline for Algorithm to test for discrete or quasi-Fuchsian subgroups of PSL(2,C)
Current License: CC BY-SA 4.0
11 events
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| Jan 6, 2022 at 7:11 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
replaced the link to the arXiv front end; see https://meta.mathoverflow.net/questions/5124/is-it-time-to-replace-links-to-the-ucdavis-arxiv-frontend
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| Oct 20, 2012 at 23:44 | comment | added | Ian Agol | @Misha: Looking at Blum-Shub-Smale, it looks like one can prove that there are degenerate groups which are not computable over R. I think this follows from the non-local connectivity results of Bromberg, together with similar arguments to section 10 of their paper. ams.org/journals/bull/1989-21-01/S0273-0979-1989-15750-9 | |
| Oct 19, 2012 at 0:00 | comment | added | Misha | @HW: I think the relevant material (with references) is in "Complexity and Real Computation", by Blum, Cucker, Shub and Smale. Of course, I could me misremembering. | |
| Oct 18, 2012 at 21:13 | comment | added | HJRW | Misha - it sounds like you're saying that algorithimic solvability implies some sort of geometric regularity. Could you give a hint or a reference for the Shashikura--Penrose material? | |
| Oct 18, 2012 at 17:36 | comment | added | Misha | ...Incidentally, showing that Hausdorff dimension of $B$ is greater than $12g-13$ is one way to prove algorithmic unsolvability (this is the argument that worked in the context of degree 2 polynomials). | |
| Oct 18, 2012 at 17:28 | comment | added | Misha | @David: At least in theory, the "thin" subset should be the boundary $B$ of the quasifuchsian space. Outside of $B$ you have an algorithm to detect membership: One machine will search for a finitely-sided fundamental domain and the other will search for pairs of group elements contradicting Jorgensen's inequality. Eventually, one of the machines stops. The set $B$ is "thin" in Baire category sense, but, I would guess, has Hausdorff dimension $12g-12$, i.e., maximal possible. Proving that Hausdorff dimension of $B$ is greater than $12g-13$, I think, is an open problem. | |
| Oct 18, 2012 at 16:37 | history | edited | Ian Agol | CC BY-SA 3.0 |
added 5 characters in body
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| Oct 18, 2012 at 13:53 | comment | added | David Dumas | Real computation is what I had in mind. Certainly I would want to allow computing words in the generators of $\rho(\Gamma)$ and comparisons between traces or matrix entries as "basic operations". Thanks for your answers, which convince me there is no hope for an algorithm in general. As in the punctured torus case, when actually implementing such a test I will need to settle for heuristics that leave a thin set in the character variety "undecided". | |
| Oct 18, 2012 at 13:27 | vote | accept | David Dumas | ||
| Oct 18, 2012 at 4:26 | comment | added | Misha | Here is another supporting piece of evidence that the discreteness problem is algorithmically unsolvable: One can ask for an algorithm to determine if a surface group representation lies in the closure of the quasifuchsian space (i.e., is discrete and faithful). This question, in the context of quadratic polynomials, is known to be algorithmically unsolvable (a theorem of Shashikura in conjunction with an observation of Penrose). There is probably enough machinery in place to prove the same result in the context of Kleinian groups. | |
| Oct 18, 2012 at 4:03 | history | answered | Ian Agol | CC BY-SA 3.0 |