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Belyi's theorem states that if a Riemann surface could be defined as an algebraic curve over an algebraic number field, then this Riemann surface could be described by a Dessin d'enfant. I have two questions:

  • Is there a way to describe the moduli space of genus $g$ bordered/punctured hyperbolic Riemann surfaces with $n$ borders/punctures using dessins d'enfants?

  • Let's say that we want to compute an integral of the form $\int_{M_{g;n}}[D\tau]\,\,f[\tau_1,\cdots,\tau_{6g-6+2n}]$ over genus $g$ bordered/punctured hyperbolic Riemann surfaces with $n$ borders/punctures in which $[D\tau]$ is a suitable measure. Mirzakhani has computed the volume of the moduli space. But what if we have a more general integral? what is the best parametrization? Fenchel-Nielsen coordinates? I suppose that it should be case-dependent.

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    $\begingroup$ 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. $\endgroup$
    – Will Sawin
    Commented May 26, 2016 at 4:31
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    $\begingroup$ 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? $\endgroup$
    – Dan Petersen
    Commented May 26, 2016 at 5:43
  • $\begingroup$ @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? $\endgroup$
    – QGravity
    Commented May 27, 2016 at 0:16
  • $\begingroup$ @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? $\endgroup$
    – QGravity
    Commented May 27, 2016 at 0:18
  • $\begingroup$ @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? $\endgroup$
    – QGravity
    Commented May 27, 2016 at 0:18

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