## Geometry & Topology

### Fundamental groups of moduli stacks of stable curves of compact type

Marco Boggi

#### Abstract

Let $ℳ˜g,n$, 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 $ℳ˜g,n$. For instance we show that the topological fundamental groups are linear, extending to all $n≥0$ previous results of Morita and Hain for $g≥2$ and $n=0,1$.

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

The Torelli group $Tg,n$ is the kernel of the natural representation $Γg,n→ Sp2g(ℤ)$. The abelianization of the Torelli group $Tg,n$ is determined for all $g≥1$ and $n≥1$, 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 $ℳ˜λ$ of $ℳ˜g,n$, for $g≥2$, has a Deligne–Mumford compactification $ℳ¯λ$ with finite fundamental group. This implies that, for $g≥3$, any finite index subgroup of $Γg$ containing $Kg$ has vanishing first cohomology group, improving a result of Hain [Math. Sci. Res. Inst. Publ. 28 (1995) 97-143].

#### Article information

Source
Geom. Topol., Volume 13, Number 1 (2009), 247-276.

Dates
Revised: 22 May 2008
Accepted: 9 September 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800180

Digital Object Identifier
doi:10.2140/gt.2009.13.247

Mathematical Reviews number (MathSciNet)
MR2469518

Zentralblatt MATH identifier
1162.32008

Keywords
Teichmüller group Torelli group

#### Citation

Boggi, Marco. Fundamental groups of moduli stacks of stable curves of compact type. Geom. Topol. 13 (2009), no. 1, 247--276. doi:10.2140/gt.2009.13.247. https://projecteuclid.org/euclid.gt/1513800180

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