Timeline for Can the limit set of an infinitely generated Schottky group have positive area?
Current License: CC BY-SA 3.0
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| Jun 1, 2017 at 19:57 | history | bounty ended | CommunityBot | ||
| May 31, 2017 at 13:43 | comment | added | Misha | @Kalim: The conjecture should hold regardless of where the limit point $p$ is. | |
| May 31, 2017 at 8:26 | comment | added | Malik Younsi | I just saw your conjectural construction, and have one question. In order to give an example of what I'm looking for, the accumulation point $p$ should belong to one of the circles $\partial D_i$, since otherwise the domain $\Omega:= S^2 \setminus \cup_i D_i$ would have a point boundary component, whereas I want only circle boundary components. Is this the situation you had in mind? | |
| May 30, 2017 at 20:23 | history | edited | Misha | CC BY-SA 3.0 |
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| May 30, 2017 at 18:14 | comment | added | Malik Younsi | It thus seems that the construction of such a domain remains open. I accepted your answer since it is very detailed and contains interesting information on relevant related constructions. Thank you! | |
| May 30, 2017 at 18:12 | vote | accept | Malik Younsi | ||
| May 26, 2017 at 20:44 | comment | added | Misha | @Kalim: It is the same construction as in Abikoff's paper. I do not believe you can modify it to suit your needs. The main thing is that the positive area comes from the boundary of a fundamental domain. This is not what you want. | |
| May 26, 2017 at 20:08 | comment | added | Malik Younsi | You mention that Abikoff's construction is useless for my purposes. I don't know if it is the construction you had in mind, but I found one construction in the book Kleinian groups and Uniformization in examples and problems, Example 36, based on Osgood's construction of a positive area Jordan curve as the closure of countably many linear segments. Packing pairwise tangent disks on those segments leads to a Schottky group whose limit set has positive area. Do you think that construction could be modified to give what I want (maybe shrinking the disks a bit to make them disjoint)? | |
| May 26, 2017 at 20:04 | comment | added | Malik Younsi | Thanks for the remarks, that is very helpful. I knew what I had in mind for a Schottky group was different than some definitions in the literature, but your answer makes the distinction much more clear. I do want the union of the closed disks to be compact. | |
| May 26, 2017 at 19:53 | history | edited | Misha | CC BY-SA 3.0 |
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| May 26, 2017 at 19:22 | history | edited | Misha | CC BY-SA 3.0 |
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| May 26, 2017 at 19:03 | history | edited | Misha | CC BY-SA 3.0 |
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| May 26, 2017 at 18:02 | comment | added | Malik Younsi | Thank you very much for the interesting references, I'll look at them closely, especially Abikoff's construction. | |
| May 25, 2017 at 17:34 | history | answered | Misha | CC BY-SA 3.0 |