# ANNALES DE L'INSTITUT FOURIER

Growth of the Weil–Petersson inradius of moduli space
Annales de l'Institut Fourier, to appear, 38 p.

In this paper we study the systole function along Weil–Petersson geodesics. We show that the square root of the systole function is uniformly Lipschitz on Teichmüller space endowed with the Weil–Petersson metric. As an application, we study the growth of the Weil–Petersson inradius of moduli space of Riemann surfaces of genus $g$ with $n$ punctures as a function of $g$ and $n$. We show that the Weil–Petersson inradius is comparable to $\sqrt{lng}$ with respect to $g$, and is comparable to $1$ with respect to $n$.

Moreover, we also study the asymptotic behavior, as $g$ goes to infinity, of the Weil–Petersson volumes of geodesic balls of finite radii in Teichmüller space. We show that they behave like $o\left({\left(\frac{1}{g}\right)}^{\left(3-ϵ\right)g}\right)$ as $g\to \infty$, where $ϵ>0$ is arbitrary.

Dans cet article, nous étudions la fonction systole le long des géodésiques de la métrique de Weil–Petersson. Nous montrons que la racine carrée de la systole est uniformément Lipschitz sur l’espace de Teichmüller muni de la métrique de Weil–Petersson. Comme application, nous étudions la croissance du rayon de la plus grande boule métrique inscrite dans l’espace des modules des surfaces de Riemann de genre $g$ avec $n$ piqûres en fonction de $g$ et $n$. Nous montrons que ce rayon est comparable à $\sqrt{lng}$ par rapport à $g$, et comparable à $1$ par rapport à $n$.

De plus, nous étudions aussi le comportement asymptotique, lorsque $g$ tends vers l’infini, des volumes de Weil–Petersson des boules géodésiques de rayons finis dans l’espace Teichmüller. Nous montrons qu’ils se comportent comme $o\left({\left(\frac{1}{g}\right)}^{\left(3-ϵ\right)g}\right)$ quand $g\to \infty$, où $ϵ>0$ est arbitraire.

Received : 2017-05-02
Revised : 2018-03-15
Accepted : 2018-05-07
Classification:  32G15,  30F60
Keywords: The moduli space, Weil–Petersson metric, inradius, large genus, systole
@unpublished{AIF_0__0_0_A36_0,
author = {Wu, Yunhui},
title = {Growth of the Weil--Petersson inradius of moduli space},
note = {to appear in \emph{Annales de l'Institut Fourier}},
}

Wu, Yunhui. Growth of the Weil–Petersson inradius of moduli space. Annales de l'Institut Fourier, to appear, 38 p.

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