Timeline for Can the limit set of an infinitely generated Schottky group have positive area?
Current License: CC BY-SA 3.0
18 events
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| S Jun 1, 2017 at 19:57 | history | bounty ended | CommunityBot | ||
| S Jun 1, 2017 at 19:57 | history | notice removed | CommunityBot | ||
| May 30, 2017 at 18:12 | vote | accept | Malik Younsi | ||
| May 27, 2017 at 15:06 | comment | added | Mirko | the following is too simplistic. You have only countably many circles, order them in a sequence, and "approximate" $\Omega$ by a sequence $\{\Omega_n\}_{n=1}^\infty$, where $\Omega_n$ is like $\Omega$ but taking only the first $n$-many circles into account (which would mean, likely, that you fill in the gaps of each circle in the remaining tail of circles). Then the idea is that you know the answer for $\Omega_n$ (zero area), and could one then use that the union of countably many sets of zero area has area zero? | |
| May 26, 2017 at 17:54 | comment | added | Malik Younsi | @IanAgol No problem! As for dimension 2, I think that Theorem 5.3 in the paper by Startmann and Urbanski cited in Igor's answer gives such an example. | |
| May 26, 2017 at 16:31 | comment | added | Ian Agol | @Karim: Okay, you're right, I was confused. There are limit sets of finitely generated Kleinian groups with Hausdorff dimension 2, but Lebesgue/Hausdorff measure 0. So I suppose a separate question is whether the Hausdorff dimension can be 2? | |
| May 25, 2017 at 20:21 | history | edited | Malik Younsi | CC BY-SA 3.0 |
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| May 25, 2017 at 20:06 | comment | added | Malik Younsi | I indeed was refer to Lebesgue measure, but aren't they proportional? | |
| May 25, 2017 at 17:34 | answer | added | Misha | timeline score: 5 | |
| May 25, 2017 at 10:14 | comment | added | YCor | "Area" certainly refers to Lebesgue measure. Hausdorff measure is natural to consider but it would be awkward to call it "area". | |
| May 24, 2017 at 22:26 | comment | added | Ian Agol | Which definition of "area" are you using? Lebesgue measure and (2-dimensional) Hausdorff measure are the natural choices. These can differ for finitely generated Kleinian groups. | |
| S May 24, 2017 at 18:35 | history | bounty started | Malik Younsi | ||
| S May 24, 2017 at 18:35 | history | notice added | Malik Younsi | Draw attention | |
| May 24, 2017 at 18:34 | history | edited | Malik Younsi | CC BY-SA 3.0 |
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| May 23, 2017 at 2:56 | answer | added | Igor Rivin | timeline score: 2 | |
| May 23, 2017 at 1:05 | history | edited | Malik Younsi | CC BY-SA 3.0 |
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| May 22, 2017 at 21:21 | answer | added | Igor Rivin | timeline score: 3 | |
| May 22, 2017 at 17:57 | history | asked | Malik Younsi | CC BY-SA 3.0 |