Triangulated Riemann Surfaces with Boundary and the Weil-petersson Poisson Structure

Given a Riemann surface with boundary S, the lengths of a maximal system of disjoint simple geodesic arcs on S that start and end at ∂S perpendicularly are coordinates on the Teichmüller space T (S). We compute the Weil-Petersson Poisson structure on T (S) in this system of coordinates and we prove that it limits pointwise to the piecewise-linear Poisson structure defined by Kontsevich on the arc complex of S. As a byproduct of the proof, we obtain a formula for the first-order variation of the distance between two closed geodesic under Fenchel-Nielsen deformation. Introduction The Teichmüller space T (S) of a Riemann surface S with punctures is endowed with a Kähler metric, first defined by Weil using Petersson’s pairing of modular forms. By the work of Wolpert ([Wol81], [Wol82] and [Wol85]), the Weil-Petersson Kähler form ωWP can be neatly rewritten using Fenchel-Nielsen coordinates. Algebraic geometers became interested in Weil-Petersson volumes of the moduli space of curves M(S) = T (S)/Γ(S) since Wolpert [Wol83a] showed that the class of ωWP is proportional to the tautological class κ1, previously defined by Mumford [Mum83] in the algebro-geometric setting and then by Morita [Mor84] in the topological setting. The reason for this interest relies on the empirical fact that many problems in enumerative geometry of algebraic curves can be reduced to the intersection theory of the so-called tautological classes (namely, ψ and κ) on the moduli space of curves. A major breakthrough in the 1980s and early 1990s was the discovery (due to Harer, Mumford, Penner and Thurston) of a cellularization of the moduli space of punctured Riemann surfaces, whose cells are indexed by ribbon graphs (also called fatgraph), that is finite graphs together with the datum of a cyclic order of the half-edges incident at each vertex. To spell it better, if S is an n-punctured Riemann surface (that is, if S = S \{x1, . . . , xn} with S compact) with χ(S) < 0, then there is a homeomorphism between M(S) × R+ and the piecewise-linear space Mcomb(S) of metrized ribbon graphs whose fattening is homeomorphic to S. By means of this cellularization, many problems could be attacked using simplicial methods (for instance, the orbifold Euler characteristic of M [HZ86] [Pen88] and the virtual homological dimension of the mapping class group Γ [Har86]). A major success was also Kontsevich’s proof [Kon92] of Witten’s conjecture [Wit91], which says that the generating series of the intersection numbers of the ψ classes on M satisfies the KdV hierarchy of partial differential equations. One of the key steps in Kontsevich’s proof was 1