# Operads of genus zero curves and the Grothendieck-Teichm\"{u}ller group

## TL;DR

This paper establishes a deep connection between the automorphisms of genus zero surface operads and the Grothendieck-Teichmüller group, revealing its nontrivial action on moduli spaces and providing an alternative proof of formality for the framed little 2-disks operad.

## Contribution

It proves the isomorphism between the homotopy automorphisms of the genus zero surface operad's profinite completion and the Grothendieck-Teichmüller group, and shows its nontrivial action on moduli spaces.

## Key findings

- Automorphisms of the genus zero surface operad correspond to the Grothendieck-Teichmüller group.
- The Grothendieck-Teichmüller group acts nontrivially on the operad of stable genus zero curves.
- An alternative proof of the formality of the framed little 2-disks operad is provided.

## Abstract

We show that the group of homotopy automorphisms of the profinite completion of the genus zero surface operad is isomorphic to the (profinite) Grothendieck-Teichm\"{u}ller group. Using a result of Drummond-Cole, we deduce that the Grothendieck-Teichm\"{u}ller group acts nontrivially on $\overline{\mathcal{M}}_{0,\bullet+1}$, the operad of stable curves of genus zero. As a second application, we give an alternative proof that the framed little 2-disks operad is formal.