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Simplicial volume of moduli spaces of Riemann surfaces. (English) Zbl 1254.30074
Summary: Motivated by results on the simplicial volume of locally symmetric spaces of finite volume, in this note, we observe that the simplicial volume of the moduli space \(M_{g,n}\) is equal to \(0\) if \(g \geq 2\); \(g = 1\), \(n \geq 3\); or \(g = 0\), \(n \geq 6\); and the orbifold simplicial volume of \(M_{g,n}\) is positive if \(g = 1\), \(n = 0, 1\); \(g = 0, n = 4\). We also observe that the simplicial volume of the Deligne-Mumford compactification of \(M_{g,n}\) is equal to \(0\), and the simplicial volumes of the reductive Borel-Serre compactification of arithmetic locally symmetric spaces \(\Gamma\backslash X\) and the Baily-Borel compactification of Hermitian arithmetic locally symmetric spaces \(\Gamma\backslash X\) are also equal to \(0\) if the \(\mathbb{Q}\)-rank of \(\Gamma\backslash X\) is at least \(3\) or if \(\Gamma\backslash X\) is irreducible and of \(\mathbb{Q}\)-rank \(2\).
MSC:
30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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