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30 1. A gentle introduction to moduli spaces of curvesThere is clearly a projection map T g,n (L) → M g,n (L) given by forgetting the marking. In fact,we obtain the moduli space as a quotient of Teichmüller space by a group action. Consider themapping class groupMod g,n = Diff + (S g,n )/Diff + 0 (S g,n),where Diff + (S g,n ) is the group of orientation preserving diffeomorphisms fixing the boundariesand Diff + 0 (S g,n) is the normal subgroup consisting of those diffeomorphisms isotopic to theidentity. There is a natural action of the mapping class group on Teichmüller space describedas follows: if [φ] is an element of Mod g,n , then [φ] sends the marked hyperbolic surface (X, f )to the marked hyperbolic surface (X, f ◦ φ).Proposition 1.25. The action of Mod g,n on T g,n (L) is properly discontinuous, though not necessarilyfree. Therefore, the quotient M g,n (L) = T g,n (L) / Mod g,n is an orbifold.Compactification and symplectificationEarlier, we described the Deligne–Mumford compactification M g,n , obtained by consideringstable algebraic curves. It should be noted that there is an analogous construction in the hyperbolicsetting, where a node of an algebraic curve corresponds to degenerating the length ofa simple closed curve on a hyperbolic surface to zero. In fact, one can construct T g,n (L), theTeichmüller space of marked stable hyperbolic surfaces, in the following way. Define a stablehyperbolic surface of type (g, n) to be a pair (X, M) where X is a surface of genus g with npunctures, M is a collection of disjoint simple closed curves on X and X \ M is endowed witha finite area hyperbolic metric. Again, we refer to a diffeomorphism f : S g,n → X as a markingand define the compactified Teichmüller space{T g,n (L) = (X, M, f )∣}/f is a marking of a stable hyperbolic surface (X, M) of∼genus g with n boundaries of lengths L 1 , L 2 , . . . , L nwhere (X, M, f ) ∼ (Y, N, g) if and only if there exists a homeomorphism φ : X → Y such thatφ(M) = N, φ restricted to X \ M is an isometry, and φ ◦ f is isotopic to g on each connectedcomponent of S g,n \ f −1 (M). Once again, the mapping class group acts on the compactifiedTeichmüller space and one may define a compactification of the moduli space M g,n (L) asM g,n (L) = T g,n (L)/Mod g,n .The moduli space M g,n (0) can be canonically identified with M g,n by the uniformisation theorem.When L ̸= 0, the moduli space M g,n (L) does not possess a natural complex structure.However, by the work of Wolpert [61], the Fenchel–Nielsen coordinates do induce a real analyticstructure. For more information on the compactification and real analytic structure ofmoduli spaces of hyperbolic surfaces, the reader is encouraged to consult the references [1, 3].
1.4. Volumes of moduli spaces 31The Teichmüller space T g,n (L) can be endowed with the canonical symplectic formω =3g−3+n∑ dl k ∧ dτ kk=1using the Fenchel–Nielsen coordinates. Although this is a rather trivial statement, a deep fact isthat this form is invariant under the action of the mapping class group. Therefore, ω descendsto a symplectic form on the quotient, namely the moduli space M g,n (L). This is referred to asthe Weil–Petersson symplectic form and we will also denote it by ω. Its existence allows us touse the techniques of symplectic geometry in the study of moduli spaces. Wolpert [61] used thereal analytic structure on M g,n (L) to show that the Weil–Petersson form extends smoothly to aclosed form on M g,n (L). In the particular case L = 0, he showed that this extension defines acohomology class [ω] ∈ H 2 (M g,n , R) which satisfies the following.Theorem 1.26. The de Rham cohomology class of the Weil–Petersson symplectic form and the characteristicclass κ 1 are related by the equation [ω] = 2π 2 κ 1 ∈ H 2 (M g,n , R).Note that for all values of L, the spaces M g,n (L) are diffeomorphic to each other, but not necessarilysymplectomorphic, when endowed with the Weil–Petersson symplectic structure. Itis therefore natural to ask how the symplectic structure varies as L varies, a topic which wediscuss in the next section.1.4 Volumes of moduli spacesWeil–Petersson volumesPowering up the Weil–Petersson symplectic form, one obtains the corresponding volume formω 3g−3+n(3g − 3 + n)! = dl 1 ∧ dτ 1 ∧ dl 2 ∧ dτ 2 ∧ . . . ∧ dl 3g−3+n ∧ dτ 3g−3+n .Of course, Teichmüller space has infinite volume with respect to this form. However, the actionof the mapping class group is such that the volume of the moduli space is finite. Thisfollows from the fact that the Weil–Petersson symplectic form can be extended smoothly to thecompactification. Therefore, define V g,n (L) to be the Weil–Petersson volume of M g,n (L). Notethat when dealing with volumes, one need not worry about the compactification of the modulispace since, by Theorem 1.2, the boundary divisor has positive codimension. In particular, itis not necessary for us to consider an extension of the Weil–Petersson symplectic form to theboundary.
Page 1: Intersection <stro
Page 4 and 5: iiDeclarationThis is to certify tha
Page 6 and 7: ivAcknowledgementsFirst and foremos
Page 8 and 9: viContentsVolumes and symplectic re
Page 10 and 11: viiiContents
Page 12 and 13: 2 0. Overviewwhich is far more trac
Page 14 and 15: 4 0. Overviewcurves. In this thesis
Page 16 and 17: 6 0. Overvieware defined to be grap
Page 18 and 19: 8 0. OverviewNote that the space of
Page 20 and 21: 10 1. A gentle introduction to modu
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Page 39: 1.3. Moduli spaces of hyperbolic su
Page 43 and 44: 1.4. Volumes of moduli spaces 33vol
Page 45 and 46: 1.4. Volumes of moduli spaces 35In
Page 47 and 48: 1.4. Volumes of moduli spaces 37reg
Page 49 and 50: 1.4. Volumes of moduli spaces 39Pro
Page 51 and 52: Chapter 2Weil-Petersson volume rela
Page 53 and 54: 2.1. Volume polynomial relations 43
Page 55 and 56: 2.2. Proofs via algebraic geometry
Page 57 and 58: 2.2. Proofs via algebraic geometry
Page 59 and 60: 2.3. Proofs via Mirzakhani’s recu
Page 69 and 70: 2.4. Applications and extensions 59
Page 75 and 76: Chapter 3A new approach to Kontsevi
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3.3. Hyperbolic surfaces with long
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3.4. Calculating the combinatorial
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Chapter 4Concluding remarksIn this
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Bibliography[1] ABIKOFF, W. The rea
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Bibliography 103[27] KRONHEIMER, P.
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Bibliography 105[58] WOLPERT, S. On
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Appendix AWeil-Petersson volumesA.1
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A.2. Program for genus 0 Weil-Peter
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