Timeline for injectivity radius of hyperbolic surface
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Apr 15, 2013 at 19:28 | comment | added | HJRW | The proof was written up by Steve D in this MO answer: mathoverflow.net/questions/9628/… . Roger Alperin wrote up an elementary proof in his paper 'An elementary account of Selberg's lemma'. | |
| Apr 15, 2013 at 16:13 | comment | added | Mikhail Katz | Do you have a reference where your remarks on Jacobson rings and linear groups are treated in more detail? | |
| Apr 15, 2013 at 15:17 | comment | added | HJRW | Re: your later comments, no, I don't think that Hempel didn't believe Thurston; rather, Thurston characteristically doesn't give all the details. Hempel's paper shows how to deduce residual finiteness from geometrization. | |
| Apr 15, 2013 at 15:12 | comment | added | HJRW | In response to your first comment, no, it's not significantly harder. In any Jacobson ring you have enough maximal ideals to prove residual finiteness. Malcev observed this in the 40s; the upshot is that any fg linear group is residually finite. | |
| Apr 15, 2013 at 14:30 | comment | added | Mikhail Katz | Hempel, John Residual finiteness for 3 -manifolds. Combinatorial group theory and topology (Alta, Utah, 1984), 379–396, Ann. of Math. Stud., 111, Princeton Univ. Press, Princeton, NJ, 1987 proved this again in the Haken case. I guess he wasn't convinced by Thurston. Does anybody know the history of this? | |
| Apr 15, 2013 at 14:23 | comment | added | Mikhail Katz | I see that the paper Thurston, William P. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381 proves RF in the Haken case. | |
| Apr 15, 2013 at 14:17 | comment | added | Mikhail Katz | Exactly my point. Congruence subgroups provide both lower bounds for systole and proof of RF for arithmetic manifolds. I don't remember if RF is also true for nonarithmetic ones, but I assume it is harder. | |
| Apr 15, 2013 at 14:12 | comment | added | HJRW | One (usually) proves residual finiteness of hyperbolic 3-manifolds using congruence quotients, so this is just the same thing. | |
| Apr 15, 2013 at 13:44 | comment | added | Mikhail Katz | I didn't find the word "congruence subgroup" on this page, so I thought I would comment that using congruence subgroups may be more elementary than appealing to residual finiteness of 3-manifolds. | |
| Apr 4, 2013 at 4:40 | vote | accept | Bidyut Sanki | ||
| Apr 4, 2013 at 4:40 | |||||
| Apr 3, 2013 at 14:51 | history | edited | Autumn Kent | CC BY-SA 3.0 |
added 86 characters in body
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| Apr 3, 2013 at 13:59 | comment | added | Misha | Richard, maybe you should also mention that you use normal finite index subgroups, this will take care of the base point business. | |
| Apr 3, 2013 at 13:35 | comment | added | Autumn Kent | Yes, I should've said that. Thanks, Misha. | |
| Apr 3, 2013 at 13:32 | comment | added | Misha | Ditto for all compact hyperbolic manifolds. | |
| Apr 3, 2013 at 11:56 | history | answered | Autumn Kent | CC BY-SA 3.0 |