Timeline for fundamental domains for free fuchsian group.
Current License: CC BY-SA 2.5
9 events
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| Apr 22, 2023 at 14:14 | comment | added | Sary | This is stated and proved on p.15 of the following paper : zbmath.org/0265.30023, which I found through this paper : zbmath.org/1095.37008. | |
| Sep 12, 2010 at 17:03 | comment | added | Benoît Kloeckner | Thanks for the answer and comments; unfortunately I found a flaw in my plan so I need to rethink all of it. References on infinite type Riemann surface are still welcome. | |
| Sep 12, 2010 at 17:01 | vote | accept | Benoît Kloeckner | ||
| Sep 11, 2010 at 18:04 | comment | added | Ian Agol | There is a canonical way to choose the geodesics. For the case of a surface of finite topology, there is a minimal convex core with totally geodesic boundary. For each proper arc, one may isotope it to be perpendicular to the boundary of the end it exits, or if the end is a cusp, one sends it to the parabolic fixed point. In the non-finite topology case, one may find a locally finite decomposition of the surface into pairs of pants. If an arc crosses infinitely many closed curves of this decomposition, one may show that there is a canonical point at infinity for the endpoint of the geodesic. | |
| Sep 11, 2010 at 5:51 | history | edited | Ian Agol | CC BY-SA 2.5 |
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| Sep 11, 2010 at 2:59 | history | edited | Ian Agol | CC BY-SA 2.5 |
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| Sep 11, 2010 at 2:29 | history | edited | Ian Agol | CC BY-SA 2.5 |
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| Sep 10, 2010 at 20:19 | comment | added | Bill Thurston | Agol's proof is correct, but it might be worth saying more about how the properly embedded topological can be replaced by geodesics. To do this, take (closed disk \ limit set) mod the fuchsian group, choose one point on each boundary component, and make all the geodesics go to that point (which comes frorm the circle at $\infty$). The corresponding statement is false in one higher dimension, for free groups in hypebolic 3-space. E.g., There are upper bounds < 2 to the Hausdorff dimension of limit sets of groups with planar fundamental domain boundaries, but the Hausdorff limit sets --> 2. | |
| Sep 10, 2010 at 18:05 | history | answered | Ian Agol | CC BY-SA 2.5 |