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 ${\Gamma }_g\to Sp(2g,\mathbb{Z})$ 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.