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 $T^{♯}$ (and its companion $T^{\ast }$) which is an extension of the Ptolemy–Thompson group $T$ by the braid group $B_{\infty }$ on infinitely many strands. We prove that $T^{♯}$ 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 $T^{♯}$ and $T^{\ast }$ are in the same relation with respect to each other as the braid groups $B_{n+1}$ 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.