Talk:Modular group

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Figures? Mtaub

I can add a lot to this...although I think the article title should just be "Modular group" rather than "Modular group Gamma", the standard notation can just be mentioned in the article itself. Also, technically, the modular group is PSL(2,Z), not SL(2,Z), although the distinction is usually glossed over in practice. There are a lot of applications of this to number theory and geometry. Revolver

Please do - I see you are working on Fuchsian groups. Eventually someone will redirect modular group here. We could have a page for Gamma-zero-of-N for all small N! (Just kidding.)

Charles Matthews 19:38, 5 Nov 2003 (UTC)

I would if I could, but I can't so I won't. (For now.) Apparently, nothing I type in will appear texed, although everything already there is fine. I went into the matrix T and changed the entries to a, b, c, and d and it WOULDN'T ******* tex it, but I put back the 1, 1, 0, 1 and it was fine again. What the hell is with that??? I'm pulling my hair out. Revolver

The problem seems to be solved. Sorry for the fuss (or maybe not, it really wasn't working yesterday.) I'll get to the other changes I wanted to yesterday in a short while. Revolver 7 Nov 2003

More to come... Revolver 7 Nov 2003

Thanks for catching that mistake with S, Gandalf. Revolver 01:09, 9 Apr 2004 (UTC)

[edit] Notation

Why the notation S*L(2,Z) for integer matrices with determinant ±1. Would it not be more clear to just write GL(2,Z)? -- Fropuff 15:37, 2005 Mar 23 (UTC)

It might be Linas who wrote that, judging from the recent contributions in the history. If this is so, and if you don't get an answer soon, I would suggest you go to Linas's talk page, as Linas has a history of not checking the watchlist too often. :) Oleg Alexandrov 15:55, 23 Mar 2005 (UTC)
S*L(2,Z) was indeed introduced on this page by Linas. But surely GL(2,Z) is the group of non-singular integer 2x2 matrices i.e. matrices with non-zero determinant ? This is not the same as what Linas means by S*L(2,Z). Having said that, I have questioned the relevance of his S*L(2,Z) to the modular group anyway, as the modular group is isomorphic to PSL(2,Z) - see this section on his talk page. Gandalf61 20:34, Mar 23, 2005 (UTC)

GL(2,Z) is just the group of integer matrices with determinant ±1. Recall that a matrix with entries in a commutative ring R is invertible iff the determinant is invertible in R. The only units in the ring of integers are ±1. Of course, if R is a field then the group of units is the set of all nonzero elements, so one recovers the usual definition. -- Fropuff 22:30, 2005 Mar 23 (UTC)

I don't log on every day, and check my watchlist only every few weeks/month... not a fast responder. Several remarks:
* There's a variety of phenomena that need to discuss the case of det=-1 to properly treat the subject. Farey sequence, fibonacci sequence, continued fractions and fractal symmetries come to mind; there are other cases that escape me. (The 2D lattice symmetries also have det=-1) Rather than starting another article to explain this case, it seemed appropriate to review the different notation in this article.
* I picked up the notation S*L from a book on Kleinian groups, which had made a point of distinguishing things. Since the book also discusses S*L(2,C) and S*L(2,R), its possible that the author chose the star notation for Z only to be consistent. We can switch the notation to GL for this article; esp. in the company of a sentance quoting Fropuff above, which is quite educational. However, if/when the articles on Kleinian/Fuchsian groups get expanded, this notational device or some variant for it will resurface.
I dont much care, as long as the result is clearer and more informative in the end. linas 15:00, 24 Mar 2005 (UTC)
As to Gandalf61's remarks about relevance, the breif answer is that I beleive that the answer is 'yes its relevant'. Here's some handwaving: the det=-1 case is needed for Farey fractions, the farey fractions number the buds on the mandelbrot set, the spectral measure of the interior of the mandelbrot set is the dedekind eta. Although the dedekind eta is a classic modular form, making reference to, and needing only PSL(2,Z), it in fact appears inside of something for which S*L(2,Z) is the more appropriate anchor. Ditto for many Kleinian/Fuchsian fractals. For the case of the discussion that you cite (regarding the link in article on Fibonacci numbers), I don't want readers who need the det=-1 case to get linked off to an article that's perfectly boring, while those who need only the det=+1 case get to link to the fascinating world of modular forms and riemann surfaces. The layman's lore about fibonacci and the golden mean is filled with superficial connections to fractals; this is not a mysterious coincidence; the explanation for the connection threads through the modular symmetries as the symmetries of "most" fractals.linas 15:26, 24 Mar 2005 (UTC)
Linas - the isomorphism between the modular group and PSL(2,Z) can be extended in a natural way to give a homomorphism from SL(2,Z) to the modular group. I don't see how you can extend this in a natural way (i.e. without making some arbitrary choices) to a homomorphism from S*L(2,Z) (a.k.a. GL(2,Z)) to the modular group. Specifically, can you explain exactly how you map a 2x2 integer matrix with determinant -1 to a Mobius transformation within the modular group ? Gandalf61 17:34, Mar 24, 2005 (UTC)
Hi Gandalf61, I'm not sure what you are trying to get at, so I don't know how to answer. If I implied that there was some homomorphism, I must have gotten overexcited during my hand-waving. I'm prone to errors when over-excited. Is this stated somewhere in the article? linas 02:27, 2 Apr 2005 (UTC)

In the arithmetic setting at least, GL(2,Z) is the standard notation for what seems to be denoted by S*L(2,Z). Furthermore these groups are of no relevance for viewpoint taken here, because for the usual action, z->az+B/cz+d, with ((ab)(cd)) in GL(2,R), exchanges the upper half plane and the lower halfplane whenever det is < 0. It is SL(2,Z) (det equal to +1) that acts on H. The viewpoint of considering the (reductive algebraic) group GL(2) instead of the (semisimple) group SL(2) or PSL(2) seems to have first, at least in higher dimensional analogues, been brought by Deligne, who was considering non connected Shimura varieties (like modular curves for instance). In this case one consider action of GL(2,Z) on the union of the two half planes, or the action of the intersection of GL(2,Z) with the connected component of GL(2,R), ie integer matrices with positive determinant, on the (connected) upper half plane. Oftenly, one then write GL(2,\mathbf{Z})^+ (which still equals SL(2,Z)), and GL(2,\mathbf{R})^+ for the connected component (with respect to the metric topology, not to be mistaken with the Zariski topology) of GL(2,R). Though the modular group is defined as the group of moebius transformations with integer coefficients, it is more natural, from the viewpoint of moduli spaces ((Deligne-Mumford) stacks more precisely) to consider the whole action of SL(2,Z) on H, though this action is trivial on the center. But maybe these remarks would be more suitable for an article on the modular curve itself. RudeWolf