Fundamental groups of moduli stacks of stable curves of compact type

Abstract

Let ${\tilde{\mathcal{M}}}_{g,n}$, 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 algebraic and topological fundamental groups of the stack ${\tilde{\mathcal{M}}}_{g,n}$. Let ${\Gamma }_{g,n}$, for $2g-2+n>0$, be the Teichmüller 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. Let $K_{g,n}$ be the normal subgroup of ${\Gamma }_{g,n}$ generated by Dehn twists along separating circles on $S_{g,n}$. As a first application of the above theory, a characterization of $K_{g,n}$ is given for all $n\ge 0$ (for $n=0,1$, this was done by Johnson). Let then ${\mathcal{T}}_{g,n}$ be the Torelli group, i.e. the kernel of the natural representation $Γ_{g,n}\ra Sp_{2g}(Z)$. The abelianization of ${\mathcal{T}}_{g,n}$ is determined for all $g\ge 1$ and $n\ge 1$, thus completing classical results by Johnson and Mess.

25 pages; minor corrections in some proofs; typos corrected; theorem numbering changed

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