The braided Ptolemy–Thompson group is asynchronously combable

Abstract

The braided Ptolemy–Thompson group $T^{\star}$ is an extension of the Thompson group $T$ by the full braid group $B_{\infty}$ on infinitely many strands and both of them can be viewed as mapping class groups of certain infinite planar surfaces. The main result of this article is that $T^{\star}$ (and in particular $T$) is asynchronously combable. The result is new already for the group $T$. The method of proof is inspired by Lee Mosher’s proof of automaticity of mapping class groups.