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

TL;DR

This paper investigates the automorphism group of an asymptotic pants complex associated with an infinite genus-zero surface, revealing its structure as an extended asymptotic mapping class group related to the braided Thompson group.

Contribution

It introduces a new automorphism group of a complex of asymptotically trivial pants decompositions and characterizes it as an extension of the braided Thompson group with additional elements.

Findings

Automorphism group is an asymptotic mapping class group.

The group is an extension of the braided Thompson group.

Provides new insights into the structure of infinite surface mapping class groups.

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.