An upper bound for the Tarski numbers of non amenable groups of piecewise projective homeomorphisms

The Tarski number of a non amenable group is the smallest number of pieces needed for a paradoxical decomposition of the group. Non amenable groups of piecewise projective homeomorphisms were introduced by Monod, and non amenable finitely presented groups of piecewise projective homeomorphisms were introduced by the author in joint work with Justin Moore. These groups do not contain non abelian free subgroups. In this article we prove that the Tarski number of all groups in both families is at most 25. In particular we demonstrate the existence of a paradoxical decomposition with 25 pieces. Our argument also applies to any group of piecewise projective homeomorphisms that contains as a subgroup the group of piecewise $PSL_2(\mathbb{Z})$ homeomorphisms of $\mathbb{R}$ with rational breakpoints, and an affine map that is a not an integer translation.