A nonamenable type F infinity group of piecewise projective homeomorphisms

摘要

We prove that the group of homeomorphisms of the circle introduced by the author with Justin Moore is of type F infinity. This makes the group the first example of a type F infinity 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.