Simplicial volume of moduli spaces of Riemann surfaces

Abstract

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 $g \ge 2$; $g = 1$, $n \ge 3$; or $g = 0$, $n \ge 6$; 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.