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

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 PSL2(Z) homeomorphisms of R with rational breakpoints and an affine map that is a not an integer translation.

Published in:
International Journal Of Algebra And Computation, 27, 3, 315-321
Singapore, World Scientific Publ Co Pte Ltd

 Record created 2017-07-10, last modified 2018-01-28

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