Timeline for Dessin d'enfant and moduli space of bordered/punctured hyperbolic Riemann surfaces
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10 events
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| May 27, 2016 at 0:46 | comment | added | Will Sawin | @QGravity Well if it's a function of the moduli space it should pull back to a function on Teichmuller space that is invariant under the mapping class group. I think if it has a simple representation, like a polynomial or simple power series, in the Fenchel-Nielsen coordinates then you're in good shape. No, I do not know how to express the period matrix or anything related in those coordinates. | |
| May 27, 2016 at 0:18 | comment | added | QGravity | @WillSawin Also do you know about any relation between Fenchel-Nielsen coordinates and elements of period matrix (at least for $g≤3$) or the action of mapping class group on the elements of period matrix? | |
| May 27, 2016 at 0:18 | comment | added | QGravity | @WillSawin could you please explain a bit more or refer me to a reference? As I understand, Mirzakhani transforms everything from moduli space to Teichmuller space using generalized MacShane identity. She expresses everything in terms of Fenchel-Nielsen coordinates. Actually I didn't understand this part of your comment : "If your function can be computed in a nice way from a pair-of-pants decomposition". Would you please explain a bit more? Do you mean it should be invariant (or well-behaved) under the action of mapping class group to be transformed to Teichmuller space? | |
| May 27, 2016 at 0:16 | comment | added | QGravity | @DanPetersen Thanks for the reference, it is quite interesting. I know about Witten's conjecture (not in great detail though!). What I am interested is to compute the integral of a function over the moduli space of Riemann surfaces. Let's say that we know the function in terms of period matrix. Mirzakhani computed the volume of moduli space. Now my question is that if there is a function in terms of coordinate of the moduli space, in which situation it is possible to integrate it explicitly and get a number? | |
| May 26, 2016 at 5:43 | comment | added | Dan Petersen | Re 1, what does exist is a well known description of the moduli space in terms of (metric) ribbon graphs, often attributed to Harer-Mumford-Penner-Thurston. And a dessin d'enfant is just a particular type of ribbon graph. A nice reference explaining this is: Mulase, M., Penkava, M.: Ribbon graphs, quadratic differentials on Riemann surfaces and algebraic curves defined over $\overline{\mathbb Q}$. Asian J. Math. 2(4), 875–920 (1998). Re 2, I'm not sure what you're asking - do you know the Witten conjecture, for instance? | |
| May 26, 2016 at 4:31 | comment | added | Will Sawin | 1. my guess is no. A dessin d'enfants is a RIemann surface with a whole lot of additional information, and it doesn't seem possible to subtract out that information to obtain a moduli space in a usual way. 2. If your function can be computed in a nice way from a pair-of-pants decomposition, then one can apply Mirzakhani's inductive method to get a formula. If not, I have no idea. | |
| S May 26, 2016 at 3:37 | history | suggested | Mark Yasuda | CC BY-SA 3.0 |
fixed minor typo in question title
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| May 26, 2016 at 3:11 | review | Suggested edits | |||
| S May 26, 2016 at 3:37 | |||||
| May 26, 2016 at 2:53 | review | First posts | |||
| May 26, 2016 at 3:10 | |||||
| May 26, 2016 at 2:53 | history | asked | QGravity | CC BY-SA 3.0 |