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 $\widehat{{\mathcal{B}}^{\frac{1}{2}}}$ of this complex is also an asymptotic mapping class group in a weaker sense. Moreover $\widehat{{\mathcal{B}}^{\frac{1}{2}}}$ is obtained by $\mathcal{B}$ by first adding new elements called half-twists and further completing it.

revised version,17p., 13 figures

References in corpus (1)

• New examples of Neuwirth-Stallings pairs and non-trivial real Milnor fibrations