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Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces. (English) Zbl 1125.30039
Let \(M_{g,n}(L_1,\dots,L_n)\) denote the moduli space of Riemann surfaces with \(n\) geodesic boundary components of lengths \(L_1,\dots,L_n\). This space carries a Weil-Petersson metric. The mapping-class group \(\mathrm{Mod}_{g,n}\) operates on it and the quotient has finite volume. The main goal of the present paper is to determine this volume. The author proves that this is given by a polynomial in \(L_1,\dots,L_n\) of total degree \(\leq 2(3g-3+n)\). Moreover if we write this as a polynomial in \(L_1/\pi,\dots,L_n/\pi\) then it is \(\pi^{6g-6+2n}\) times a polynomial with rational coefficients which can be computed explicitly. The author gives some examples. This generalizes results of S. Wolpert and P. Zograf.
This remarkable formula is based on an identity due to G. McShane [Invent. Math. 132, No. 3, 607–632 (1998; Zbl 0916.30039)] and of a generalization thereof, also proved in this paper. The formula is in a certain sense an integrated version of this identity.

30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14H81 Relationships between algebraic curves and physics
Full Text: DOI
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