The theory of Big Mapping Class groups is still in its infancy, but it is beginning to emerge as a serious field of investigation after recent seminal work of J. Bavard and others. One of the most interesting applications promises to be to 1-dimensional complex dynamics, which is both a rich source of examples, and a testing ground for the power of the abstract machinery being developed.
This course aims to explain some of the background and theory of both big mapping class groups and complex dynamics, and to study some specific examples of the interaction in some depth.
Notes for this class will be written and posted online as we progress. The final version of the notes aims to be a self-contained monograph. Current version of the notes can be downloaded here.