Talk:Teichmüller space

New article

I've started developing the article on this topic. I'm fairly new to the subject, however, and I've only seen a limited number of presentations of the material. I'd like suggestions (and contributions, of course) on making this article as accessible and useful as possible. JPB 04:03, 4 September 2005 (UTC)

In the line "... surface X (or its underlying topological structure) provides a marking X → Y of each Riemann surface Y represented in TX..." perhaps there is a link you can give to help me understand what you mean by marking. Dewa (talk) 19:28, 13 May 2008 (UTC)

Should add Beltrami coefficient to define Teichmuller spacec as B(X)_1/Diff_0(X) Wolfraine (talk) 10:55, 6 December 2009 (UTC)

Perhaps it would be helpful if the page included a definition. --70.233.154.17 (talk) 00:47, 13 March 2010 (UTC)

Annulus?

If you remove from the sphere two simply connected domains, you get something conformally equivalent to an annulus. So, these things, up to conformal equivalence, are characterized by the conformal modulus of the annulus, one real parameter. The article seems to say the Teichmuller space is infinite complex dimensional, because we have disks removed, pieces of ideal boundary. But these conformal equivalences seem to be isotopic to the identity... so, I'm not sure what the reason is for the difference between one real and infinite complex dimensions. Maybe some silly convention about fixing the maps on the boundary pieces? And, assuming the Teichmuller space is indeed considered infinite dim, is the space of annuli up to conformal equivalence a well-established space? Say, the moduli space of complex annuli, or I don't know how to call this? The moduli space article is not very readable for me, unfortunately. --GaborPete (talk) 08:23, 24 February 2010 (UTC)

Explanation of topology and complex structure missing

The article explains T_X as a set, but not as a topological space or as a complex manifold. This is a pretty big omission. AxelBoldt (talk) 06:07, 19 November 2012 (UTC)

Rewrite

I completely rewrote most of the article, adding some new stuff. The historical section is not good, it could use a rigorous going-over in particular w.r.t. sourcing. I have put two "Main" templates that redirect to the page itself, as a reminder that the topics concerned should have their own page (this one is in my opinion of the correct length, give or take a few more sections on topics not yet included). jraimbau (talk) 09:50, 8 July 2016 (UTC)

There is still one "Main" template that redirects here (in the Teichmüller metric section). jraimbau (talk) 11:09, 2 August 2016 (UTC)

Teichmüller Metric?

Why isn't there more on the Teichmüller metric, only the tiny stub? Apparently there used to be a separate article?

Also, is there a reason there are no citations to any of Teichmüller's papers? I know several were published in very inaccessible places, but they've since been republished, and things like publication dates don't appear in the article. --Dylan Thurston (talk) 18:32, 1 November 2017 (UTC)

• I could not find a trace of a proper article on the Teichmüller metric (both Teichmüller metric and Teichmüller mapping redirect here and their history is blank). Obviously the section on this topic could be more developed, and I think a separate article would be the best place to put it as this one is already long enough. Note that there are already some statements on the large-scale properties of the Teichmüller metric in a later section. jraimbau (talk) 14:28, 2 November 2017 (UTC)

Teichmüller space of the torus: PSL(2, Z)?

Isn't the Teichmüller space of the torus $H^2/PSL(2, Z)$? --91.59.107.32 (talk) 18:50, 24 November 2021 (UTC)

No, that's the moduli space of elliptic curves. Teichmüller space is the moduli space of marked elliptic curves: a point \tau in H^2 gives an elliptic curve C/(Z + Z\tau) together with a diffeomorphism to a fixed topological torus (say the latter is a square with opposite sides glued, then the diffeo sends the segments [0,1], [0, \tau] to the sides). Points that differ by a transformation by an element of PSL_2(Z) correspond to the same elliptic curve but the diffeomorphisms are not isotopic to each other (for example because they act differently on homology). In other words the points in Teichmüller space are different but their images in moduli space are the same. jraimbau (talk) 15:47, 25 November 2021 (UTC)