Geometry & Topology
- Geom. Topol.
- Volume 13, Number 1 (2009), 247-276.
Fundamental groups of moduli stacks of stable curves of compact type
Let , for , be the moduli stack of –pointed, genus , stable complex curves of compact type. Various characterizations and properties are obtained of both the topological and algebraic fundamental groups of the stack . For instance we show that the topological fundamental groups are linear, extending to all previous results of Morita and Hain for and .
Let , for , be the Teichmüller group associated with a compact Riemann surface of genus with points removed , ie the group of homotopy classes of diffeomorphisms of which preserve the orientation of and a given order of its punctures. Let be the normal subgroup of generated by Dehn twists along separating simple closed curves (briefly s.c.c.) on . The above theory yields a characterization of for all , improving Johnson’s classical results for closed and one-punctured surfaces in [Topology 24 (1985) 113-126].
The Torelli group is the kernel of the natural representation . The abelianization of the Torelli group is determined for all and , 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 of , for , has a Deligne–Mumford compactification with finite fundamental group. This implies that, for , any finite index subgroup of containing has vanishing first cohomology group, improving a result of Hain [Math. Sci. Res. Inst. Publ. 28 (1995) 97-143].
Geom. Topol., Volume 13, Number 1 (2009), 247-276.
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
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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