Talk:Fuchsian group

I'd like to see the pronnouncation of the word Fuchsian (in several languages) thx User:mtaub

It should be pronounced as english "fooksian", but with a short "oo" (as the "u" in German "Bus" or "Schluss").--129.70.14.127 (talk) 05:09, 24 February 2008 (UTC)

a crystal clear definition would be nice

I'm guessing from this page that a Fuchsian group is a torsion-free discrete subgroup of PSL(2,R). It would be nice to make this crystal clear. Right now it says a Fuchsian group is a discontinuous subgroup of PSL(2,R), then goes on to define what this means... and then it says a Fuchsian group is also torsion free. The naive reader might think this is a fact about discontinuous subgroups of PSL(2,R), rather than another clause in the definition of "Fuchsian subgroup". Since I know a bunch of discontinuous subgroups that aren't torsion free, I guess "torsion free" must be part of the definition.

However, the page gives the modular group PSL(2,Z) as an example of a Fuchsian group, and the modular group is not torsion free.

So, I'm confused. John Baez 05:48, 9 May 2005 (UTC)

I doubt torsion-free belongs in the definition. Yes, discrete subgroup in precisely what Lie group should be clarified in the intro. Morally I suppose it should be in GL₂(R), modulo its maximal compact subgroup. I think what is trying to be said is, acts discontinuously on the upper half-plane. We have a convention difficulty about GL, SL, PGL and PSL which I'm not sure is sorted out yet. Charles Matthews 11:11, 9 May 2005 (UTC)

My fault for adding 'torsion-free'; I can't imagine what I was thinking. I was skimming a book on the topic at the time, and it seemed to make sense in some strange way :-/ As I'm interested in the general area, I should know this stuff; I'll volunteer to go to the library and re-study-up, but it might take a few weeks.

Whowever wrote the first sentance seemed to be trying to imply that discrete subgroups of general Lie groups can also be called "fuchsian". I have no idea if that's true; it "sounds nice", though. If I find a reference for this, I'll add it. linas 23:59, 9 May 2005 (UTC)

Well, no, Fuchsian group is specific, as is Kleinian group, Picard group. The lattice (group) definition has an advantage, namely it says the quotient space G/Γ has finite invariant measure; and this is something that passing to a quotient by a compact subgroup respects. So it equivalent to saying fundamental domain of finite measure, with respect to the right measure (i.e. y⁻²dxdy in the upper half-plane). Charles Matthews 08:38, 10 May 2005 (UTC)

I see from another encyclopedia that PSL₂(R) is used; and that fundamental domain of finite measure defines Fuchsian group of the first kind in classical terminology. Charles Matthews 21:59, 11 May 2005 (UTC)

I do promise to go to the library and look this up. It will just take me a while, in part because I lost my library card when I fell out of a canoe :) linas 23:40, 11 May 2005 (UTC)

Well, for now I am going to edit the page to say that a Fuchsian group is a discrete subgroup G of PSL(2,R), since that's what they say here:

D. Long and W. Reid, On Fuchsian groups with the same set of axes

and also here:

Juha Hataaja, Discrete groups and the Dirichlet polygon

It's sort of ridiculous that I haven't gotten ahold of any more authoritative source, like Svetlana Katok's book Fuchsian Groups, because I'm actually interested in this subject now. But, this is the second time I've been looking around on the web for info on Fuchsian groups and run across this Wikipedia article, so I'm going to correct it as well as I can for now.

I'm mildly puzzled by the idea that a Fuchsian group is just any discrete subgroup of PSL(2,R), because this would include finite subgroups, even the trivial group. The idea of having a fundamental domain with finite area in the hyperbolic plane sounds good. But, for now I'll go with the above authors!

As for PSL versus PGL versus SL versus GL, I think this is a problem built into the whole subject which will never be completely resolved: all 4 groups are interesting and important and slightly different. For example, one of the above authors says that a Fuchsian group is a discrete subgroup of PSL(2,R), but that he will also use this term for a discrete subgroup of SL(2,R).

But, of these four, PSL(2,R) seems the one most used in this context: SL(2,R) and GL(2,R) aren't subgroups of the isometries of the hyperbolic plane, while PGL(2,R) includes orientation-reversing transformations. Only PSL(2,R) is the group of orientation-preserving isometries of the hyperbolic plane, and this is the one that people like here... for example, the moduli space of elliptic curves is H/PSL(2,Z) John Baez 17:36, 30 May 2005 (UTC)

As it happens, one of my library books is recalled, so I have to go to the library today. There was an excellent book on Fuchsian groups there, but some student had nabbed it before I did. Now that school's out, maybe I'll get a shot at it. It wasn't Svetlana's book, I remember this one being head & shoulders better. John you're not at UT, and recalling my subriemannian geometry book, are you? linas 14:41, 1 Jun 2005 (UTC)

OK, started an expansion and hopefully clarification; more to come. I'm taking a break. Unfortunately, finite cyclic groups also count as fuchsian. The torsion free comment applies to the fundamental domain, I'll try to fit that in correctly, shortly. linas 18:05, 5 Jun 2005 (UTC)

Fuchsian or Kleinian?

The current version of the article mentions PSL(2,C). Now as far as I know, subgroups of the latter are called Kleinian groups, not Fuchsian groups. If so, there is no need to mention PSL(2,C) on this page. Katzmik 12:35, 5 August 2007 (UTC)

A Fuchsian group is a particular kind of subgroup of PSL(2,C). One common definition is that it is the image of a discrete, faithful representation of a surface group such that its limit set is a geometric circle. You can conjugate such a representation so that the image is acting on the standard H^2 included in H^3. So then it is a subgroup of PSL(2,R). The definitions in the article about the action on the sphere at infinity are equivalent to this one. One might wonder why not just stick with PSL(2, R), but there is the whole theory of quasi-Fuchsian groups, which are deformations of Fuchsian groups. I gather that in the modern theory after Thurston one really needs to study all this stuff as happening in H^3 and use the 3-manifold theory. —The preceding unsigned comment was added by 24.10.30.15 (talk) 06:35, August 21, 2007 (UTC)

Example

Could you give an example of a Fuchsian group of second type (other than the trivial group)? Thanks.--129.70.14.127 (talk) 05:04, 24 February 2008 (UTC)

This is not my area, but I think perhaps there is very little difference between the two (except geometrically, which might be what people care about). (Vinberg 2001) says that "Every finitely-generated Fuchsian group of the second kind is topologically isomorphic (as a group of the disc) to a finitely-generated Fuchsian group of the first kind", so I suspect a group of the second kind is just conjugate to the group of the first kind by an element of PSL(2,C) that does not preserve the real line.

• Vinberg, E.B. (2001) [1994], "Fuchsian Group", Encyclopedia of Mathematics, EMS Press

Hope it helps, and if so, you might include an explicit example in the article. JackSchmidt (talk) 06:01, 24 February 2008 (UTC)