Fundamental groups of moduli stacks of stable curves of compact type

Boggi, Marco

doi:10.2140/gt.2009.13.247

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Fundamental groups of moduli stacks of stable curves of compact type

Marco Boggi

Geometry & Topology 13 (2009) 247–276

DOI: 10.2140/gt.2009.13.247

Abstract

Let ${\tilde{ℳ}}_{g, n}$ , for $2 g - 2 + n > 0$ , 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 ${\tilde{ℳ}}_{g, n}$ . For instance we show that the topological fundamental groups are linear, extending to all $n \geq 0$ previous results of Morita and Hain for $g \geq 2$ and $n = 0, 1$ .

Let $Γ_{g, n}$ , for $2 g - 2 + n > 0$ , 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 $K_{g, n}$ be the normal subgroup of $Γ_{g, n}$ generated by Dehn twists along separating simple closed curves (briefly s.c.c.) on $S_{g, n}$ . The above theory yields a characterization of $K_{g, n}$ for all $n \geq 0$ , improving Johnson’s classical results for closed and one-punctured surfaces in [Topology 24 (1985) 113-126].

The Torelli group $T_{g, n}$ is the kernel of the natural representation $Γ_{g, n} \to {Sp}_{2 g} (ℤ)$ . The abelianization of the Torelli group $T_{g, n}$ is determined for all $g \geq 1$ and $n \geq 1$ , 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 ${\tilde{ℳ}}^{λ}$ of ${\tilde{ℳ}}_{g, n}$ , for $g \geq 2$ , has a Deligne–Mumford compactification ${\bar{ℳ}}^{λ}$ with finite fundamental group. This implies that, for $g \geq 3$ , any finite index subgroup of $Γ_{g}$ containing $K_{g}$ 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

References

Publication

Received: 4 December 2007

Revised: 22 May 2008

Accepted: 9 September 2008

Preview posted: 3 November 2008

Published: 1 January 2009

Proposed: Shigeyuki Morita

Seconded: Joan Birman, Walter Neumann

Authors

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

Volume 13, issue 1 (2009)