Abstract:
Let $S$ be an oriented closed surface of genus $g$. The mapping class group of $S$ is the group of orientation preserving homeomorphisms of $S$ onto itself considered up to isotopy. Theory of mapping class groups is closely related to geometry and topology of moduli spaces, topology of three-dimensional manifolds, braid groups, and automorphisms of free groups. The action of the mapping class group on the first homology group of the surface $S$ yields a surjective homomorphism of it to the arithmetic group $\mathrm{Sp}(2g,\mathbb{Z})$. The kernel of this homomorphism is called the Torelli group and denoted by $\mathcal{I}_g$; it is the most mysterious part of the mapping class group. It is well known that the group $\mathcal{I}_1$ is trivial. Mess (1992) proved that $\mathcal{I}_2$ is an infinitely generated free group. On the other hand, Johnson (1983) showed that $\mathcal{I}_g$ is finitely generated, provided that $g>2$. One of the most interesting questions concerning Torelli groups is whether the groups $\mathcal{I}_g$ are finitely presented or not for $g>2$. This question is closely related to the problem of computing the homology of the Torelli groups. In the talk I will give a survey of recent results on the homology of Torelli groups, focusing on a possible approach to proving that $\mathcal{I}_3$ is not finitely presented. The main tool is the study of the action of $I_g$ on a contractible CW complex constructed by Bestvina, Bux, and Margalit (2007) and called the complex of cycles.