Timeline for Automorphism groups of cocompact Fuchsian groups as mapping class groups
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Oct 26, 2018 at 14:44 | comment | added | AGenevois | In the book Surfaces and Planar Discontinuous Groups, written by Heiner Zieschang, Elmar Vogt and Hans-Dieter Coldewey, I found an entire chapter dedicated to automorphism groups of planar groups. The result I am looking for does not seem to be written there, but it seems to be good starting point. | |
| Oct 19, 2018 at 15:09 | comment | added | Lee Mosher | Having said all of that, I don't know of a reference. One place I might look is in work of Zieschang, who cleaned up lots of little nooks in surface theory over the years. But I cannot say whether this search will be fruitful. | |
| Oct 19, 2018 at 15:04 | comment | added | Lee Mosher | Triangle groups (and other groups such as polygon reflection groups) do not really fit into this question, for a few reasons. First, a reflection isometry of $\mathbb H^2$ reverses orientation, whereas Fuchsian groups are discrete subgroups of $\text{PSL}(2,\mathbb R)$ acting as the group of orientation preserving isometries of $\mathbb H^2$. Second, while one can formulate this question using the full group of isometries, the presentations have a different form from those in the question: they need relations associated to finite dihedral subgroups where two reflection lines cross at a point. | |
| Oct 18, 2018 at 22:27 | comment | added | AGenevois | Right. (One possibility to justify that the Out of a triangle group is finite is to notice that any triangle group satisfies Serre's property FA and next to apply Paulin's theorem.) In this case, the Out should be virtually isomorphic to $\mathrm{Mod}(S_{0,3})$, but $\mathrm{Mod}(S_{0,4})$ seems to be a virtually free group, so definitely not commensurable to a triangle group. I will try to understand what happens in this case. | |
| Oct 18, 2018 at 21:06 | comment | added | YCor | The Out of some cocompact Fuchsian groups such as triangle groups is finite, isn't it? | |
| Oct 18, 2018 at 20:56 | history | edited | YCor | CC BY-SA 4.0 |
fixed typo; edited tags
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| Oct 18, 2018 at 20:47 | history | edited | AGenevois | CC BY-SA 4.0 |
Corrected spelling
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| Oct 18, 2018 at 11:32 | history | asked | AGenevois | CC BY-SA 4.0 |