  Pursueing our investigations on the relations between Thompson groups and
mapping class groups, we introduce the group $T^*$ (and its further
generalizations) which is an extension of the Ptolemy-Thompson group $T$ by
means of the full braid group $B_{\infty}$ on infinitely many strands. We prove
that it is a finitely presented group with solvable word problem, and give an
explicit presentation of it.
