  We prove that the group of homeomorphisms of the circle introduced by the
author with Justin Moore (Groups, Geometry and Dynamics 2015) is of type
$F_{\infty}$. This makes the group the first example of a type $F_{\infty}$
group which is nonamenable and does not contain nonabelian free subgroups. To
prove our result we provide a certain generalisation of cube complexes, which
we refer to as cluster complexes. We also obtain a computable normal form, or a
canonical unique choice of a word for each element of the group.
