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Various characterizations and properties are obtained of both the algebraic and topological fundamental groups of the stack $\\widetilde{\\cal M}_{g,n}$.\n  Let $\\Gamma_{g,n}$, for $2g-2+n>0$, be the Teichm\\\"uller group associated with a compact Riemann surface of genus $g$ with $n$ points removed $S_{g,n}$, i.e. 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. 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Various characterizations and properties are obtained of both the algebraic and topological fundamental groups of the stack $\\widetilde{\\cal M}_{g,n}$.\n  Let $\\Gamma_{g,n}$, for $2g-2+n>0$, be the Teichm\\\"uller group associated with a compact Riemann surface of genus $g$ with $n$ points removed $S_{g,n}$, i.e. 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. 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