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)
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 (which still equals SL(2,Z)), and 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
This section doesn't seem to be correct.
- The dyadic monoid
seems to be a term for the monoid generated by all finite strings of symbols A and B with A2 = B2 = 1: this happens to be the free product of groups .So the definition isn't correct. The modular group, on the other hand, contains the free product , which isn't the same thing as either of the above.
- (Later, not quite right, see below)
- I don't understand what it means to say "This monoid occurs naturally in the study of fractal curves, and describes the self-similarity symmetries of the Cantor function, Minkowski's question mark function, and the Koch curve, each being a special case of the general de Rham curve." Firstly, how does it "describe" anything? Secondly, the Cantor function is not a curve. Should that read "the graph of the Cantor function"?
- "The monoid also has higher-dimensional linear representations". Does this mean a map into a general linear group?
- "the N=3 representation can be understood ..." not by me I'm afraid, since this is the first mention of N in the section. Is it a reference back to the previous on congruence subgroups?
- Sorry, got that first point a bit wrong. The dyadic monoid is the free monoid on 2 generators, ie all finite strings of the form XXX...YYY...XXX...YYY... . What's described in the article is the monoid on generators S and T subject to the relation S2 = something, possibly 1. The modular group contains the free product of cyclic groups , which isn't the same thing as either of those two. However, because that free product contains free groups of all finite rank, in particular it contains, but is not equal to, the free group on two generators, which in turn contains but is not equal to, the free monoid on two generators. Richard Pinch (talk) 23:01, 15 July 2008 (UTC)
- I think this section would be a lot saner if S and T were specified. There are countably many copies of the free monoid on two generators in PSL(2,Z), and if any copy will do, then I don't see why it is described here, rather than simply linking to the Tits alternative. I suspect that its relation to the fractals is somewhat interesting, perhaps being generators of the set of endomorphisms of the fractal? Do we know who wrote this section? It might be polite to drop them a note. WP:V suggests taking a look at the index of the only reference given. If "dyadic monoid" is in there, then we should try to fix the section. JackSchmidt (talk) 01:55, 16 July 2008 (UTC)
- That last diff was interesting, as it explains the origin of the stray N=3 that I didn't understand. Just to record that text here: "The modular group also has pairs of representations in GL(N,R). The N=3 representation can be understood to describe the self-symmetry of the blancmange curve, and higher N 's as the symmetries of similar curves." I'm not sure it's correct but it certainly makes more sense. However, it seems to be in essence about the monoid and perhaps should go to Free monoid if anywhere. Richard Pinch (talk) 06:08, 16 July 2008 (UTC)
Revert experimental markup changes
I have reverted a series of experimental markup and formatting changes that were made (and then partly reversed) in a series of edits by User:Yecril. I feel strongly that significant and non-standard markup changes like this should only be made after consensus has been sought and gained and after obvious bugs in the new markup have been ironed out, so I am reverting these changes pending a wider discussion within the Wikipedia community. I have suggested to Yecril that he should initiate a discussion of his ideas at Wikipedia talk:WikiProject Mathematics, but so far he seems to be reluctant to do this - see thread at User talk:Yecril. Gandalf61 (talk) 08:37, 4 September 2008 (UTC)
Hey, I made a visualization showing how Klein's j-invariant is invariant under action of SL(2,R).
<a href="http://www.youtube.com/watch?v=kvQ7e2AiE2g">Klein's j-Invariant j(τ) under τ→-1/τ in SL(2,R)</a>
If anyone wants to turn this into an animation for this page, I would be happy to supply them with the movie file for it.