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Fundamental groups of moduli stacks of stable curves of compact type

Marco Boggi

Geometry & Topology 13 (2009) 247–276

Let ˜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 ˜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(). 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 ˜λ of ˜g,n, for g 2, has a Deligne–Mumford compactification ¯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].

Teichmüller group, Torelli group
Mathematical Subject Classification 2000
Primary: 32G15
Secondary: 14H10, 30F60, 14F35
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
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