Abstract
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].
Citation
Marco Boggi. "Fundamental groups of moduli stacks of stable curves of compact type." Geom. Topol. 13 (1) 247 - 276, 2009. https://doi.org/10.2140/gt.2009.13.247
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