## Journal of Differential Geometry

### Growth of Weil-Petersson Volumes and Random Hyperbolic Surface of Large Genus

Maryam Mirzakhani

#### Abstract

In this paper, we investigate the geometric properties of random hyperbolic surfaces of large genus. We describe the relationship between the behavior of lengths of simple closed geodesics on a hyperbolic surface and properties of the moduli space of such surfaces. First, we study the asymptotic behavior of Weil-Petersson volume $V^{g,n}$ of the moduli spaces of hyperbolic surfaces of genus $g$ with $n$ punctures as $g \to \infty$. Then we discuss basic geometric properties of a random hyperbolic surface of genus $g$ with respect to the Weil-Petersson measure as $g \to \infty$.

#### Article information

Source
J. Differential Geom., Volume 94, Number 2 (2013), 267-300.

Dates
First available in Project Euclid: 1 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1367438650

Digital Object Identifier
doi:10.4310/jdg/1367438650

Mathematical Reviews number (MathSciNet)
MR3080483

Zentralblatt MATH identifier
1270.30014

#### Citation

Mirzakhani, Maryam. Growth of Weil-Petersson Volumes and Random Hyperbolic Surface of Large Genus. J. Differential Geom. 94 (2013), no. 2, 267--300. doi:10.4310/jdg/1367438650. https://projecteuclid.org/euclid.jdg/1367438650