Аннотация:
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 Sp(2g,Z). The kernel of this homomorphism is called the Torelli group and denoted by Ig; it is the most mysterious part of the mapping class group. It is well known that the group I1 is trivial. Mess (1992) proved that I2 is an infinitely generated free group. On the other hand, Johnson (1983) showed that Ig is finitely generated, provided that g>2. One of the most interesting questions concerning Torelli groups is whether the groups Ig 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 I3 is not finitely presented. The main tool is the study of the action of Ig on a contractible CW complex constructed by Bestvina, Bux, and Margalit (2007) and called the complex of cycles.