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

Abstract

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.

72 Pages. Referee comments have been incorporated. To appear in the Journal of Topology

References in corpus (2)

• Groups of piecewise projective homeomorphisms
• A finitely presented group of piecewise projective homeomorphisms

Cited by in corpus (3)

• Non-triviality of the Poisson boundary of random walks on the group $H(\mathbb{Z})$ of Monod
• The BNSR-invariants of the Lodha-Moore groups, and an exotic simple group of type ${\text{F}}_{\infty }$
• Finiteness Properties of Locally Defined Groups