PROBLEMS, QUESTIONS, AND CONJECTURES ABOUT MAPPING CLASS GROUPS

We discuss a number of open problems about mapping class groups of surfaces. In particular, we discuss problems related to linearity, congruence subgroups, cohomology, pseudoAnosov stretch factors, Torelli groups, and normal subgroups. Beginning with the work of Max Dehn a century ago, the subject of mapping class groups has become a central topic in mathematics. It enjoys deep and varied connections to many other subjects, such as lowdimensional topology, geometric group theory, dynamics, Teichmüller theory, algebraic geometry, and number theory. The number of papers on mapping class groups recorded on MathSciNet in the last six decades has grown from 205 to 386 to 525 to 791 to 1,121 to 1,390. The subject seems to enjoy an endless supply of beautiful ideas, pictures, and theorems. At the 2017 Georgia International Topology Conference, the author gave a lecture called “Problems and progress on mapping class groups.” This paper is a summary of parts of that lecture. What follows is not a comprehensive list in any way, even among the topics it attempts to address. Rather it gives a mix of problems—from the famous and notoriously difficult to the eminently doable. There is little attempt to give background; the reader may find that in the book by Farb and the author [59] and in the other references therein. There are other problem lists on mapping class groups. In fact Farb has edited an entire book of problem lists on mapping class groups [53]. Ivanov’s problem list [81], which also appears in Farb’s book, has been particularly influential on the author of this article. That problem list is an updated version of a problem list written in conjunction with the 1993 Georgia International Topology Conference, twenty-four years earlier. Some of Ivanov’s problems also appeared in Kirby’s problem This material is based upon work supported by the National Science Foundation under Grant No. DMS 1057874.