The braided Ptolemy-Thompson group is asynchronously combable

The braided Ptolemy-Thompson group T-star is an extension of the Thompson group T by the full braid group B-infinity 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.