Timeline for Is every Cantor set $C\subseteq\mathbb R^{\infty}$ the limit set of a Fuchsian group?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 8 at 20:41 | vote | accept | Christian Remling | ||
| Oct 8 at 20:34 | comment | added | Daniel Asimov | No, it is complerely unclear, because that notation is virtually never used with that meaning. But that notation is often used for two other meanings I am aware of: a) the increasing union of all ℝ^n, and b) the countable cartesian power of ℝ. | |
| Oct 8 at 20:28 | comment | added | Christian Remling | $\mathbb R^{\infty}=\mathbb R \cup \{ \infty \}$, which should be clear enough from the context. | |
| Oct 8 at 20:11 | comment | added | YCor | I'm also confused by the notation $\mathbf{R}^\infty$, and know the notations $\mathrm{P}^1_\mathbf{R}$, $\mathrm{P}^1(\mathbf{R})$, $\mathbf{R}\mathrm{P}^1$. | |
| Oct 8 at 20:09 | history | edited | YCor | CC BY-SA 4.0 |
moved question to body, added tag
|
| Oct 8 at 18:12 | answer | added | Moishe Kohan | timeline score: 5 | |
| Oct 8 at 17:46 | history | edited | Christian Remling | CC BY-SA 4.0 |
added 390 characters in body
|
| Oct 8 at 17:41 | comment | added | Moishe Kohan | The best reference I know is Nicholls "Ergodic theory of discrete groups." For "other" obstructions, I would have to think, I am sure there are many, just less obvious. A stupid example would be to take two Fuchsian Schottky groups with different critical exponents and disjoint limit sets. Then take the union of their limit sets. | |
| Oct 8 at 17:39 | comment | added | Christian Remling | @MoisheKohan: Also, are there other "well known" properties of $C$? Though I normally hate this kind of thing, I might adapt the question accordingly... | |
| Oct 8 at 17:38 | comment | added | Christian Remling | @MoisheKohan: No. Do you have a reader friendly reference for this? | |
| Oct 8 at 17:34 | comment | added | Moishe Kohan | Do you know that limit set of any nonelementary Kleinian group has positive Hausdorff dimension? Incidentally, the notation $\mathbb R^\infty$ is normally used for a certain infinite-dimensional real vector space. The usual notation for the boundary of the hyperbolic plane is different. | |
| Oct 8 at 17:26 | comment | added | Christian Remling | @MoisheKohan: I'm not sure the hint is helping me. Could you be a bit more explicit please. | |
| Oct 8 at 17:25 | comment | added | Moishe Kohan | Hint: Consider a Cantor subset of zero Hausdorff dimension. | |
| Oct 8 at 17:16 | history | asked | Christian Remling | CC BY-SA 4.0 |