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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)
##### Keywords:
Riemann surface; moduli space; simplicial volume
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