Fundamental groups of moduli stacks of stable curves of compact type

Let M̃g,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 algebraic and topological fundamental groups of the stack M̃g,n. 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, i.e. 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 circles on Sg,n. As a first application of the above theory, a characterization of Kg,n is given for all n ≥ 0 (for n = 0, 1, this was done by Johnson in [J3]). We define the Torelli group Tg,n to be the kernel of the natural representation Γg,n → Sp2g(Z). The abelianization of the Torelli group Tg,n is determined for all g ≥ 1 and n ≥ 1, thus completing classical results by Johnson [J4] and Mess [Me] (for a different definition of the Torelli group, this was done by van den Berg in [vdB], who provides also a new proof of Johnson’s and Mess’ results). Mathematics Subject Classifications (2000): 14H10, 30F60, 14F35, 14H15, 32G15.