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- Title
An upper bound for the Tarski numbers of nonamenable groups of piecewise projective homeomorphisms.
- Authors
Lodha, Yash
- Abstract
The Tarski number of a nonamenable group is the smallest number of pieces needed for a paradoxical decomposition of the group. Nonamenable groups of piecewise projective homeomorphisms were introduced in [N. Monod, Groups of piecewise projective homeomorphisms, Proc. Natl. Acad. Sci. 110(12) (2013) 4524-4527], and nonamenable finitely presented groups of piecewise projective homeomorphisms were introduced in [Y. Lodha and J. T. Moore, A finitely presented non amenable group of piecewise projective homeomorphisms, Groups, Geom. Dyn. 10(1) (2016) 177-200]. These groups do not contain non-abelian free subgroups. In this paper, 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 homeomorphisms of with rational breakpoints and an affine map that is a not an integer translation.
- Subjects
FPF rings; FREE groups; PIECEWISE affine systems; PROJECTIVE geometry; TORSION free Abelian groups
- Publication
International Journal of Algebra & Computation, 2017, Vol 27, Issue 3, p315
- ISSN
0218-1967
- Publication type
Article
- DOI
10.1142/S0218196717500151