On the automorphisms group of the asymptotic pants complex of an infinite surface of genus zero

Abstract

The braided Thompson group $\mathcal B$ is an asymptotic mapping class group of a sphere punctured along the standard Cantor set, endowed with a rigid structure. Inspired from the case of finite type surfaces we consider a Hatcher-Thurston cell complex whose vertices are asymptotically trivial pants decompositions. We prove that the automorphism group $\hat{\mathcal B^{\frac{1}{2}}}$ of this complex is also an asymptotic mapping class group in a weaker sense. Moreover $\hat{\mathcal B^{\frac{1}{2}}}$ is obtained by $\mathcal B$ by first adding new elements called half-twists and further completing it.