Timeline for Can the limit set of an infinitely generated Schottky group have positive area?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| May 26, 2017 at 18:43 | comment | added | Misha | @Kalim: It is due to Abikoff. | |
| May 26, 2017 at 18:05 | comment | added | Malik Younsi | @LasseRempe-Gillen I'm not sure I understand the construction that you have in mind. Could you give a bit more details? Thanks :) | |
| May 26, 2017 at 14:50 | comment | added | Lasse Rempe | @Kalim It should not matter whether the discs are disjoint or not in my construction. Just make them a little bit smaller than in the packing, but closer and closer to it as you go to the boundary. | |
| May 24, 2017 at 21:04 | comment | added | Igor Rivin | @LasseRempe-Gillen See the edit - I did what I had suggested. | |
| May 24, 2017 at 21:04 | history | edited | Igor Rivin | CC BY-SA 3.0 |
gave Rich Schwartz' example.
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| May 24, 2017 at 20:48 | comment | added | Malik Younsi | @LasseRempe-Gillen Note that I want an example of a circle domain (closed disks are disjoint), not a circle packings (open disks are disjoint). I seem to recall seeing somewhere the construction of an infinite chain of tangent disks separating the plane whose limit set is a given Jordan curve of positive area. I can't remember where though... | |
| May 24, 2017 at 14:55 | comment | added | Lasse Rempe | @IgorRivin Fair point! :) | |
| May 24, 2017 at 14:53 | comment | added | Igor Rivin | @LasseRempe-Gillen Yes it does, one can write to Rich and ask :) | |
| May 24, 2017 at 11:17 | comment | added | Lasse Rempe | In the paper, he writes "We remark that it is well known that one can produce (infinitely generated) Schottky groups whose limit sets have positive volume". Of course, this doesn't help with finding a reference. But, can't you build an example by hand, just make denser and denser packings with tiny radii as you approach some point of the boundary? | |
| May 23, 2017 at 1:07 | comment | added | Malik Younsi | Also, note that Schwartz studies packings, where only the interior of the disks are assumed to be disjoint. In particular, limit sets of even finitely generated packings can be Jordan curves. I, on the other hand, assume that the closed disks are disjoint, and that the complement of the union of the closed disks is both open and connected. The structure of the limit set is very different in this case. | |
| May 23, 2017 at 1:06 | comment | added | Malik Younsi | Ah I was careless, sorry. Of course the limit set can have zero area... I meant to ask if the limit set can have positive area. I edited accordingly, thanks! | |
| May 22, 2017 at 21:21 | history | answered | Igor Rivin | CC BY-SA 3.0 |