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

A. A. Gaifullin

Izvestiya. Mathematics , Tome 85 (2021) no. 6, pp. 1060-1127

Résumé

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 of all mapping classes that act trivially on the homology of $S_g$. One of the most intriguing open 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 second homology group of $\mathcal{I}_3$ using the spectral sequence $E^r_{p,q}$ for the action of $\mathcal{I}_3$ on the complex of cycles. In this paper we obtain evidence for the 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 $E^3_{0,2}$ of the spectral sequence is not finitely generated, that is, the group $E^1_{0,2}$ remains infinitely 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 image of $d^3$ would complete the proof that $\mathcal{I}_3$ is not finitely presented.

Détail

Keywords: Torelli group, mapping class group, homology of groups, complex of cycles, action of a group on a complex, spectral sequence, Birman–Craggs homomorphisms.