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 monograph aims to explain some of the background and the abstract theory of both of these subjects, and to focus on a few specific examples where the interaction is particularly deep and fruitful.
The version available here for download is pre- whatever the letter is that comes before alpha in the Greek alphabet. Most sections are incomplete or entirely unwritten. It corresponds roughly to material I hope to cover in a graduate course I will teach at the University of Chicago Winter 2019.
Download .pdf (preliminary version 5/2/2019)