A modular operad of mapping class groups

The action of the absolute Galois group of the rationals on the little disks operad has several important implications, including for the study of the embedding tower of knots, as shown by Boavida de Brito–Horel.
In this talk, we follow Grothendieck’s idea that such Galois group should also act on mapping class groups of all surfaces with boundary, in a way that is compatible with surface inclusions. We discuss an operadic model for such a higher-genus Teichmüller tower. To this end, we introduce a new modular operad in groupoids designed to encode key features of the “Lego–Teichmüller game” and show that the profinite completion of this modular operad admits an action of the absolute Galois group of Q.