The Grothendieck-Teichmüller group is an explicitly defined group introduced by Drinfel'd which is closely related to (and conjecturally equal to) the absolute Galois group. The idea was based on Grothendieck's suggestion that one should study the absolute Galois group of the rationals by relating it to its action on the Teichmüller tower of fundamental groupoids of the moduli stacks of genus g curves with n marked points.
In this talk, we give a reimagining of the genus zero Teichmüller tower in terms of a profinite completion of the framed little 2-discs operad. Using this reinterpretation, we show that the homotopy automorphisms of this model for the Teichmüller tower are isomorphic to the (profinite) Grothendieck-Teichmüller group. We then show a non-trivial action of the absolute Galois group on our tower.
This talk will be aimed at a general audience and will not assume previous knowledge of the Grothendieck-Teichmüller group or operads. This is joint work with Pedro Boavida and Geoffroy Horel.