Fundamental groups of moduli stacks of stable curves of compact type

Details Fundamental groups of moduli stacks of stable curves of compact type Fundamental groups of moduli stacks of stable curves of compact type

2006-04-12

arxiv.org/abs/math/0604271v2

Mathematics

Algebraic Geometry

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

Scientific paper

Let $\widetilde{\cal M}_{g,n}$, 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 $\widetilde{\cal M}_{g,n}$. 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. 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 $n\geq 0$ (for $n=0,1$, this was done by Johnson). Let then ${\cal T}_{g,n}$ be the Torelli group, i.e. the kernel of the natural representation $\Gamma_{g,n}\ra Sp_{2g}(Z)$. The abelianization of ${\cal T}_{g,n}$ is determined for all $g\geq 1$ and $n\geq 1$, thus completing classical results by Johnson and Mess.

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