Fundamental groups of moduli stacks of stable curves of compact type

Boggi, Marco¹

¹ Escuela de Matemática, Universidad de Costa Rica, Ciudad Universitaria Rodrigo Facio, San Pedro de Montes de Oca, Apartado 2060, San José, Costa Rica

Geometry & topology, Tome 13 (2009) no. 1, pp. 247-276.

Résumé

Let ${\widetilde{\mathcal{ℳ}}}_{g,n}$, for $2g-2+n>0$, be the moduli stack of $n$–pointed, genus $g$, stable complex curves of compact type. Various characterizations and properties are obtained of both the topological and algebraic fundamental groups of the stack ${\widetilde{\mathcal{ℳ}}}_{g,n}$. For instance we show that the topological fundamental groups are linear, extending to all $n\ge 0$ previous results of Morita and Hain for $g\ge 2$ and $n=0,1$.

Let ${\Gamma }_{g,n}$, for $2g-2+n>0$, be the Teichmüller group associated with a compact Riemann surface of genus $g$ with $n$ points removed $S_{g,n}$, ie the group of homotopy classes of diffeomorphisms of $S_{g,n}$ which preserve the orientation of $S_{g,n}$ and a given order of its punctures. Let ${\mathcal{K}}_{g,n}$ be the normal subgroup of ${\Gamma }_{g,n}$ generated by Dehn twists along separating simple closed curves (briefly s.c.c.) on $S_{g,n}$. The above theory yields a characterization of ${\mathcal{K}}_{g,n}$ for all $n\ge 0$, improving Johnson’s classical results for closed and one-punctured surfaces in [Topology 24 (1985) 113-126].

The Torelli group ${\mathcal{T}}_{g,n}$ is the kernel of the natural representation ${\Gamma }_{g,n}\to {Sp}_{2g}\left(\mathbb{Z}\right)$. The abelianization of the Torelli group ${\mathcal{T}}_{g,n}$ is determined for all $g\ge 1$ and $n\ge 1$, thus completing classical results of Johnson [Topology 24 (1985) 127-144] and Mess [Topology 31 (1992) 775-790] for closed and one-punctured surfaces.

We also prove that a connected finite étale cover ${\widetilde{\mathcal{ℳ}}}^{\lambda }$ of ${\widetilde{\mathcal{ℳ}}}_{g,n}$, for $g\ge 2$, has a Deligne–Mumford compactification ${\overline{\mathcal{ℳ}}}^{\lambda }$ with finite fundamental group. This implies that, for $g\ge 3$, any finite index subgroup of ${\Gamma }_g$ containing ${\mathcal{K}}_g$ has vanishing first cohomology group, improving a result of Hain [Math. Sci. Res. Inst. Publ. 28 (1995) 97-143].

Détail

DOI : 10.2140/gt.2009.13.247

Keywords: Teichmüller group, Torelli group

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