#### Volume 13, issue 1 (2009)

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Fundamental groups of moduli stacks of stable curves of compact type

### Marco Boggi

Geometry & Topology 13 (2009) 247–276
##### Abstract

Let ${\stackrel{˜}{\mathsc{ℳ}}}_{g,n}$, 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 ${\stackrel{˜}{\mathsc{ℳ}}}_{g,n}$. 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 ${\mathsc{K}}_{g,n}$ 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 ${\mathsc{K}}_{g,n}$ 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 ${\mathsc{T}}_{g,n}$ is the kernel of the natural representation ${\Gamma }_{g,n}\to {Sp}_{2g}\left(ℤ\right)$. The abelianization of the Torelli group ${\mathsc{T}}_{g,n}$ 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 ${\stackrel{˜}{\mathsc{ℳ}}}^{\lambda }$ of ${\stackrel{˜}{\mathsc{ℳ}}}_{g,n}$, for $g\ge 2$, has a Deligne–Mumford compactification ${\overline{\mathsc{ℳ}}}^{\lambda }$ with finite fundamental group. This implies that, for $g\ge 3$, any finite index subgroup of ${\Gamma }_{g}$ containing ${\mathsc{K}}_{g}$ has vanishing first cohomology group, improving a result of Hain [Math. Sci. Res. Inst. Publ. 28 (1995) 97-143].

##### Keywords
Teichmüller group, Torelli group
##### Mathematical Subject Classification 2000
Primary: 32G15
Secondary: 14H10, 30F60, 14F35