  The braided Ptolemy-Thompson group $T^*$ is an extension of the Thompson
group $T$ by the full braid group $B_{\infty}$ on infinitely many strands. This
group is a simplified version of the acyclic extension considered by Greenberg
and Sergiescu, and can be viewed as a mapping class group of a certain infinite
planar surface. In a previous paper we showed that $T^*$ is finitely presented.
Our main result here is that $T^*$ (and $T$) is asynchronously combable. The
method of proof is inspired by Lee Mosher's proof of automaticity of mapping
class groups.
