Geometry & Topology

Fundamental groups of moduli stacks of stable curves of compact type

Marco Boggi

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Abstract

Let ˜g,n, for 2g2+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 ˜g,n. For instance we show that the topological fundamental groups are linear, extending to all n0 previous results of Morita and Hain for g2 and n=0,1.

Let Γg,n, for 2g2+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 n0, 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(). The abelianization of the Torelli group Tg,n is determined for all g1 and n1, 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 ˜λ of ˜g,n, for g2, has a Deligne–Mumford compactification ¯λ with finite fundamental group. This implies that, for g3, 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
Received: 4 December 2007
Revised: 22 May 2008
Accepted: 9 September 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
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

Subjects
Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
Secondary: 14H10: Families, moduli (algebraic) 30F60: Teichmüller theory [See also 32G15] 14F35: Homotopy theory; fundamental groups [See also 14H30]

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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