Let M_(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 M_(g,n). For instance, in Theorem 3.20, 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 velence 0,1. Let (GAMMA)_(g,n), for 2g - 2 + n > 0, be the Teichmuller group associated with a compact Riemann surface of genus g with n points removed S_(g,n), ie the group of homotopy classes of diffeomorphisms of S_(g,n) which preserve the orientation of S_(g,n) and a given order of its punctures. Let (kappa)_(g,n) be the normal subgroup of (GAMMA)_(g,n) generated by Dehn twists along separating simple closed curves (briefly s.c.c.) on S_(g,n). The above theory yields a characterization of (kappa)_(g,n) for all n >= 0, improving Johnson's classical results for closed and one-punctured surfaces in [13]. The Torelli group T_(g,n) is the kernel of the natural representation (GAMMA)_(g,n) -> Sp_(2g) (Z). The abelianization of the Torelli group T_(g,n) is determined for all g >= 1 and n >= 1, thus completing classical results of Johnson [14] and Mess [18] for closed and one-punctured surfaces. We also prove that a connected finite etale cover M~(lambda) of M_(g,n), for g >= 2, has a Deligne-Mumford compactification M~(lambda) with finite fundamental group. This implies that, for g >= 3, any finite index subgroup of (GAMMA)_(g) containing (kappa)_(g) has vanishing first cohomology group, improving a result of Hain [8].
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