On a spectral sequence for the action of the Torelli group of genus $3$ on the complex of cycles

The Torelli group of a closed oriented surface $S_g$ of genus $g$ is the subgroup$\mathcal{I}_g$ of the mapping class group $\operatorname{Mod}(S_g)$ consisting ofall mapping classes that act trivially on the homology of $S_g$. One of the most intriguingopen problems concerning Torelli groups is the question of whether the group $\mathcal{I}_3$is finitely presented. A possible approach to this problem relies on the study of the secondhomology group of $\mathcal{I}_3$ using the spectral sequence $E^r_{p,q}$ for the actionof $\mathcal{I}_3$ on the complex of cycles. In this paper we obtain evidence forthe conjecture that $H_2(\mathcal{I}_3;\mathbb{Z})$ is not finitely generated and hence$\mathcal{I}_3$ is not finitely presented. Namely, we prove that the term $E^3_{0,2}$ ofthe spectral sequence is not finitely generated, that is, the group $E^1_{0,2}$ remainsinfinitely generated after taking quotients by the images of the differentials $d^1$ and $d^2$.Proving that it remains infinitely generated after taking the quotient by the imageof $d^3$ would complete the proof that $\mathcal{I}_3$ is not finitely presented.

Torelli group, mapping class group, homology of groups, complex of cycles, action of a group ona complex, spectral sequence, Birman–Craggs homomorphisms