Torelli group

The Torelli group $I_g$ is the subgroup of the mapping class group $\Gamma_g$ that acts trivially on the first homology of the surface $F_g$. It is the kernel of the map $\Gamma_g \rightarrow Sp(2g,\mathbb{Z})$ and captures (in some sense) the difference between the topology of $\mathcal{M}_g$ and that of $\mathcal{A}_g$, the moduli space of principally polarized abelian varieties