Finite generation and algebraicity of homology of Torelli groups

The mapping class group of an oriented closed surface $S_g$ of genus $g$ is the group $\mathrm{Mod} \left( S_g \right)$ of all orientation-preserving homeomorphisms of $S_g$ considered up to isotopy. This group also admits an equivalent purely algebraic definition: $\mathrm{Mod} \left( S_g \right)$ is an index $2$ subgroup of the outer automorphism group $\mathrm{Out} \left( \pi_1 \left( S_g \right) \right)$. The most important subgroup of $\mathrm{Mod} \left( S_g \right)$ is the Torelli group $\mathcal{I}_g$, consisting of all mapping classes acting trivially on the first homology of $S_g$, i.e., on the abelianization of $\pi_1 \left( S_g \right)$. Johnson (1983) proved that $\mathcal{I}_g$ is finitely generated, provided that $g \geqslant 3$. An old open question is whether the group $\mathcal{I}_g$ is finitely presented. Another related problem is whether the homology groups $H_k \left( \mathcal{I}_g \right)$ are finitely generated in some stable range of degrees $k \ll g$ tending to infinity as $g \to \infty$. We will give a positive solution to th latter problem. The key ingredient is Tavgen's result on the bounded generation of the group $\mathrm{Sp}_{2g} \left( \mathbb{Z} \right)$ for $g \geqslant 2$.