Intersection theory on moduli spaces of curves ... - User Web Pages

7that one obtains what is referred to as a metric ribbon graph. We refer to this numberas the length of the edge and the sum of the numbers around a boundary as the lengthof the boundary. The space of metric ribbon graphs of type (g, n) with boundary lengthsprescribed by x = (x 1 , x 2 , . . . , x n ) forms a topological space — in fact, an orbifold — whichwe denote by MRG g,n (x). Using an argument analogous to that of Bowditch and Epsteinwhich appears in [5], we prove the following.Theorem 3.9. The spaces M g,n (x) and MRG g,n (x) are homeomorphic as orbifolds.Therefore, one may equivalently consider MRG g,n (x) rather than M g,n (x). One advantageis that the space of metric ribbon graphs possesses the natural orbifold cell decompositionMRG g,n (x) =⋃MRG Γ (x),Γ∈RG g,nwhere a metric ribbon graph lies in the set MRG Γ (x) if its underlying ribbon graph coincideswith Γ. Furthermore, MRG Γ (x) is top-dimensional if and only if the ribbon graphΓ is trivalent. Since the volume does not care about cells of positive codimension, it canbe expressed as a sum over trivalent ribbon graphs as follows, where V Γ (x) denotes theWeil–Petersson volume of MRG Γ (x).V g,n (x) =∑Γ∈TRG g,nV Γ (x)Part 2: Why do we obtain a product over the edges of the trivalent ribbon graph?Fix a ribbon graph Γ ∈ TRG g,n and an n-tuple of positive real numbers x = (x 1 , x 2 , . . . , x n ).Then for every positive real number N, we have the mapf : MRG Γ (x) → MRG Γ (Nx) → M g,n (Nx),which is a homeomorphism onto its image. This is the composition of two maps — thefirst scales the ribbon graph metric by a factor of N while the second uses the Bowditch–Epstein construction. Consider the normalised Weil–Petersson symplectic form ω onN 2M g,n (Nx) and note that it pulls back via f to a symplectic form on MRG Γ (x). The behaviourof this 2-form as N approaches infinity is described by the theorem below. Theproof of this statement, which relies on a mixture of hyperbolic geometry and combinatorics,is one of the main technical contributions in this part of the thesis. The key geometricintuition involved is the fact that, as the boundary lengths of a hyperbolic surfaceapproach infinity, the surface resembles a ribbon graph after appropriate rescaling of themetric. It has been brought to our attention that this result, with an alternative proof, hasalso appeared recently in the work of Mondello [36].Theorem 3.20. In the N → ∞ limit, f ∗ ωN 2 converges pointwise on MRG Γ (x) to a 2-form Ω.