No finite index subgroup of a mapping class group embeds into the $C^2$ circle diffeomorphism group

One intriguing direction of research in surface theory is the analogy between mapping class groups and higher rank lattices. However, current knowledge on finite index subgroups of mapping class groups are still scarce. We prove the result in the title, which was originally asked by Farb, and which is analogous to Ghys and Burger–Monod theorem on obstructions of higher rank lattice actions on the circle. (Joint work with Hyungryul Baik and Thomas Koberda)