Fundamental groups of moduli stacks of stable curves of compact type

Marco Boggi

Published 2006-04-12, updated 2007-01-11Version 2

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.