A type $F_{\infty}$ group of piecewise projective homeomorphisms

The purpose of this article is to prove that the group of homeomorphisms of the circle introduced in arXiv:1308.4250v3 is of type $F_{\infty}$. This provides the first example of a type $F_{\infty}$ group which is nonamenable and does not contain nonabelian free subgroups. We also obtain a normal form, or a canonical unique choice of words for the elements of the group.