Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces
type: Item
instance of: scholarly article
title: Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces
zbMATH Open document ID: 1125.30039
author: Maryam Mirzakhani
published in: Inventiones Mathematicae
publication date: 2007-01-26T00:00:00Z
review text: 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 \textit{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.
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 14H81
zbMATH DE Number: 5118879
zbMATH Keywords: Weil-Petersson metric
zbMATH Keywords: Teichmüller space
zbMATH Keywords: Teichmüller modular group
zbMATH Keywords: bordered Riemann surfaces
zbMATH Keywords: McShane's identity
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
full work available at URL: s00222-006-0013-2
OpenAlex ID: W1986699862
Wikidata QID: Q33083249
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DOI: 10.1007/S00222-006-0013-2
description: scientific article; zbMATH DE number 5118879