Thompson's group $T$ is the orientation-preserving automorphism group of a cellular complex

Abstract

We consider a planar surface $\Sigma$ of infinite type which has Thompson's group $\mathcal{T}$ as asymptotic mapping class group. We construct the asymptotic pants complex $\mathcal{C}$ of $\Sigma$ and prove that the group $\mathcal{T}$ acts transitively by automorphisms on it. Finally, we establish that the automorphism group of the complex $\mathcal{C}$ is an extension of the Thompson group $\mathcal{T}$ by $\mathbb{Z}/2\mathbb{Z}$