Finite generation, algebraicity, and representation stability for homology of Torelli groups

Abstract

We solve a long-standing problem of whether the homology groups of the Torelli subgroups $\mathcal{I}_g\le\mathrm{Mod}_g$ are finitely generated in stable range. Namely, we prove that the group $H_k(\mathcal{I}_g;\mathbb{Z})$ is finitely generated, provided that $k\le g-2$. Two main ingredients of our approach are as follows. First, we show that the action of any symplectic transvection $t_x\in\mathrm{Sp}_{2g}(\mathbb{Z})$ on the homology of $\mathcal{I}_g$ satisfies the following unipotency condition: $(t_x-1)^{k+1}H_k( \mathcal{I}_g;\mathbb{Z})=0$. The proof of this fact relies on the study of the spectral sequence for the action of $\mathcal{I}_g$ on the complex of homologous curves on $\Sigma_g$. The second key ingredient is Tavgen's theorem asserting that the group $\mathrm{Sp}_{2g}(\mathbb{Z})$ is boundedly elementarily generated. For homology with coefficients in $\mathbb{Q}$, we further prove that $H_k(\mathcal{I}_g;\mathbb{Q})$ is an algebraic $\mathrm{Sp}_{2g}(\mathbb{Z})$-representation in the same stable range $k\le g-2$. Kupers and Randal-Williams have obtained a conditional result: they computed the algebraic part of the rational cohomology of Torelli groups in stable range under the assumpition that the rational cohomology groups are finite-dimensional in this stable range. Our results turn this conditional computation into a precise theorem that describes the whole rational cohomology ring of Torelli groups in stable range. As further applications, we, firstly, prove Morita's conjecture asserting that the $\mathrm{Sp}_{2g}(\mathbb{Z})$-invariant part of the rational cohomology of $\mathcal{I}_g$ stabilizes to the polynomial ring $\mathbb{Q}[e_2,e_4,\ldots]$ in the even Miller-Morita-Mumford classes; secondly, we prove the uniform representation stability for the series of groups $\left\{ H_k\bigl(\mathcal{I}_g^1;\mathbb{Q})\right\}_{g=1}^{\infty}$.