Fifteen Problems about the Mapping Class Groups

Let S be a compact orientable surface, possibly with boundary. We denote by ModS the mapping class group of S, i.e. the group π0(Diff(S)) of isotopy classes of diffeomorphisms of S. This group is also known as the Teichmüller modular group of S, whence the notation ModS. Note that we include the isotopy classes of orientation-reversing diffeomorphisms in ModS, so our group ModS is what is sometimes called the extended mapping class group. For any property of discrete groups one may ask if ModS has this property. Guidance is provided by the analogy, well-established by now, between the mapping class groups and arithmetic groups. See, for example, [I1] or [I7] for a discussion. After the 1993 Georgia Topology Conference, R. Kirby was preparing a new version of his famous problem list in low-dimensional topology. In response to his appeal, I prepared a list of ten problems about the mapping class groups, which was informally circulated as a preprint [I5]. Some of these problems ended up in Kirby’s list [Ki] in a somewhat modified form, some did not. In the present article I will indicate the current status of these problems and also will present some additional problems. Only Problems 4,6, and 9 of the original list of ten problems are by now completely solved. (For the convenience of the readers familiar with [I5] I preserved the numbering of these ten problems.) I tried to single out some specific questions, leaving aside such well-known problems as the existence of finite dimensional faithful linear representations or the computation of the cohomology groups.