12 problems

Looijenga's conjecture. The smooth Torelli group is trivial:

Folklore conjecture. For each $k$, there exists some $G_k$ such that for $g \geq G_k$, the homology group $\operatorname{H}_k(\operatorname{\mathcal{I}}_g;\mathbb{Q})$ is finite-di…

Let $\mathcal{I}_{g,1}$ be the Torelli group of a genus-$g$ surface with one boundary component, let $\operatorname{IA}_{2g}$ be the IA-automorphism group of a free group of rank…

Let $\mathcal{I}_g$, $\mathcal{I}_{g,1}$ and $\mathcal{I}_g^1$ be the Torelli groups of a closed surface, a surface with one boundary component, and a surface with one marked point…

Let $\mathcal{I}_g$ be the Torelli group of a closed genus-$g$ surface, and let $e_{2i}$ denote the Mumford–Morita–Miller class of degree $4i$. Kawazumi–Morita's conjecture. Stably…

Let $\mathcal{I}_g$, $\mathcal{I}_{g,1}$ and $\mathcal{I}_g^1$ be the Torelli groups of a closed surface, a surface with one boundary component, and a surface with one marked point…

Let $S$ be a surface and let $H$ be a nonelementary subgroup of its mapping class group. A cover $\widetilde{S}$ to which $H$ lifts is a cover on which every element of $H$ admits…

Let $\mathcal{I}_{n,1}$ be the Torelli group of a genus-$n$ surface with one boundary component and let $\operatorname{IA}_n$ be the IA-automorphism group of the free group of rank…

Let $\mathcal{I}_{n,1}$ be the Torelli group of a genus-$n$ surface with one boundary component and let $\operatorname{IA}_n$ be the IA-automorphism group of the free group $F_n$.…

Let $\mathcal{I}_{g,1}$ be the Torelli group of a genus-$g$ surface with one boundary component, and let $\operatorname{gr}(\mathcal{I}_{g,1})$ be the graded rational Lie algebra a…

Stable lower central series conjecture. The lower central series of the Torelli group stably coincides with the restriction of the $Y$-filtration:

Torelli group commuting-or-free conjecture. Either $g$ and $h$ commute, or the subgroup they generate is a free group: