## Journal of Differential Geometry

### Simplicial volume of moduli spaces of Riemann surfaces

Lizhen Ji

#### 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.

#### Article information

Source
J. Differential Geom., Volume 90, Number 3 (2012), 413-437.

Dates
First available in Project Euclid: 24 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1335273390

Digital Object Identifier
doi:10.4310/jdg/1335273390

Mathematical Reviews number (MathSciNet)
MR2916042

Zentralblatt MATH identifier
1254.30074

#### Citation

Ji, Lizhen. Simplicial volume of moduli spaces of Riemann surfaces. J. Differential Geom. 90 (2012), no. 3, 413--437. doi:10.4310/jdg/1335273390. https://projecteuclid.org/euclid.jdg/1335273390