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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeMar 22nd 2014
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexStubby beginning for [[Moebius transformation]].

   Stubby beginning for Moebius transformation.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 22nd 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks. I have added some more cross-links.

   Thanks.

   I have added some more cross-links.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeMar 22nd 2014
   • (edited Mar 22nd 2014)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThanks, Urs. I added some more material, but I must have made some subtle syntax error, because for some reason it's not rendering as it should. Perhaps I'm tired. Edit: fixed. I still have no idea what I did wrong, but I did somehow fix whatever it was.

   Thanks, Urs. I added some more material, but I must have made some subtle syntax error, because for some reason it’s not rendering as it should. Perhaps I’m tired.

   Edit: fixed. I still have no idea what I did wrong, but I did somehow fix whatever it was.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 5th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexI made _[[modular group]]_ redirect to _[[Möbius transformation]]_ and added some more cross-links (also with [[moduli stack of elliptic curves]] etc.) This used to redirect to _[[modular theory]]_.

   I made modular group redirect to Möbius transformation and added some more cross-links (also with moduli stack of elliptic curves etc.) This used to redirect to modular theory.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 5th 2014
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI added material on the modular group to [[Moebius transformation]], but I see now there's plenty of overlap with [[moduli stack of elliptic curves]]. That's okay I guess.

   I added material on the modular group to Moebius transformation, but I see now there’s plenty of overlap with moduli stack of elliptic curves. That’s okay I guess.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeApr 5th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks!! (There is currently some syntax error that confuses the parser in the section "Action on hyperbolic space". I would try to fix it, but I have to dash off right now...)

   Thanks!!

   (There is currently some syntax error that confuses the parser in the section “Action on hyperbolic space”. I would try to fix it, but I have to dash off right now…)

7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeApr 6th 2014
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexCan we generalise from fields to, say, integral domains? Since at the beginning of the article $k$ is a field, then soon after we have $k=\mathbb{Z}$.

   Can we generalise from fields to, say, integral domains? Since at the beginning of the article $k$ is a field, then soon after we have $k=\mathbb{Z}$.

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 6th 2014
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI think we can generalize to commutative rings. When it was first written it was for fields, and then later Urs began adding a bit on the modular group, which caused consideration of $k = \mathbb{Z}$. It shouldn't surprise me a bit of people consider these things for adeles as well, which aren't even integral domains.

   I think we can generalize to commutative rings. When it was first written it was for fields, and then later Urs began adding a bit on the modular group, which caused consideration of $k = \mathbb{Z}$. It shouldn’t surprise me a bit of people consider these things for adeles as well, which aren’t even integral domains.

9. • CommentRowNumber9.
   • CommentAuthorDavidRoberts
   • CommentTimeApr 7th 2014
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI restricted to integral domains because I was wondering for what rings it makes sense to talk about the projective line. Clearly $PSL_2(R)$ makes perfect sense for any ring $R$.

   I restricted to integral domains because I was wondering for what rings it makes sense to talk about the projective line. Clearly $PSL_2(R)$ makes perfect sense for any ring $R$.

10. • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 7th 2014
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexIt's a good question. We have an entry [[projective space]] which gives a general definition, but there's also a scheme-theoretic notion of projective line (hence a functor $CRing \to Set$) and it's not immediately obvious to me what the relation is between these notions. I suppose there ought to be on general principle a Klein-geometry approach as well, where we are first given a concrete group of transformations, and we extract a geometry from what it is the group action preserves (e.g., it could be cross-ratio or something like it).

    It’s a good question. We have an entry projective space which gives a general definition, but there’s also a scheme-theoretic notion of projective line (hence a functor $CRing \to Set$) and it’s not immediately obvious to me what the relation is between these notions. I suppose there ought to be on general principle a Klein-geometry approach as well, where we are first given a concrete group of transformations, and we extract a geometry from what it is the group action preserves (e.g., it could be cross-ratio or something like it).