The braided Ptolemy-Thompson group is finitely presented

Abstract

Pursuing our investigations on the relations between Thompson groups and mapping class groups, we introduce the group T {music sharp sign} (and its companion T*) which is an extension of the Ptolemy -Thompson group T by the braid group B ∞ on infinitely many strands. We prove that T {music sharp sign} is a finitely presented group by constructing a complex on which it acts cocompactly with finitely presented stabilizers, and derive from it an explicit presentation. The groups T {music sharp sign} and T * are in the same relation with respect to each other as the braid groups Bn+1 and B n, for infinitely many strands n. We show that both groups embed as groups of homeomorphisms of the circle and their word problem is solvable. © 2008 Mathematical Sciences Publishers.