Timeline for Fuchsian groups and their normalizers
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 4, 2013 at 18:56 | vote | accept | expmat | ||
| Feb 27, 2013 at 23:45 | answer | added | Misha | timeline score: 8 | |
| Feb 27, 2013 at 22:48 | comment | added | François Brunault | An obvious case is $\Gamma=\operatorname{PSL}_2(\mathbf{Z})$ : in this case $N(\Gamma)/\Gamma$ is trivial while the quotient $\Gamma \backslash \mathcal{H}$ is isomorphic to $\mathbf{C}$, whose automorphism group is much larger. | |
| Feb 27, 2013 at 22:45 | comment | added | expmat | @Misha: can you give me a reference for those things? | |
| Feb 27, 2013 at 21:42 | comment | added | Misha | In general, the relation (for nonelementary Fuchsian groups $\Gamma$) is: $N(\Gamma)< Aut(\Gamma)$ (since such $\Gamma$ has trivial centralizer in $PSL(2,R)$) and $N(\Gamma)/\Gamma< Aut(H^2/\Gamma)$. If you treat the quotient $H^2/\Gamma$ as an orbifold (and its best if you do), then the latter inclusion is the equality. | |
| Feb 27, 2013 at 20:50 | comment | added | expmat | Or some other (more complicated) relation that holds in general for any $\Gamma$? | |
| Feb 27, 2013 at 20:42 | comment | added | expmat | Right, but can we at least say that $N(\Gamma)$ is a subgroup of $Aut(\Gamma)$? | |
| Feb 27, 2013 at 20:34 | comment | added | Venkataramana | It can happen, I think, that $\Gamma \backslash {\mathcal H}$ may even be the projective line, if $\Gamma$ has torsion; in that case, the automorphisms of the Riemann surface is $PGL(2, {\mathbb C})$, much larger than $N(\Gamma)$. | |
| Feb 27, 2013 at 20:30 | history | edited | expmat | CC BY-SA 3.0 |
added 47 characters in body
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| Feb 27, 2013 at 20:29 | comment | added | expmat | Yes, but what about the case where $\Gamma$ is not torsion free? | |
| Feb 27, 2013 at 20:20 | comment | added | Venkataramana | If $\Gamma$ is torsion free, then they are the same. Any automorphism of the Riemann surface, by the lifting criterion, lifts to an automorphism of the upper half plane (i.e. an element of $SL(2,{\mathbb R})$ . This element normalises $\Gamma$ almost by construction. | |
| Feb 27, 2013 at 20:15 | history | asked | expmat | CC BY-SA 3.0 |