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

The braided Thompson group 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 groupB 1 2 ̂ of this complex is also an asymptotic mapping class group in a weaker sense. MoreoverB 1 2 ̂ is obtained by B by first adding new elements called half‐twists and further completing it.