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 [Formula: see text] homeomorphisms of [Formula: see text] with rational breakpoints and an affine map that is a not an integer translation.

ON CONFIGURATION GRAPH AND PARADOXICAL DECOMPOSITION

In this paper, we introduce the concept of configuration graph and show how one can use this notion to simplify the theorem proved by Rejali and Yousofzadeh [Configuration of groups and paradoxical decompositions, Bull. Belg. Math. Soc. Simon Stevin18 (2011) 157–172].

A continuous movement version of the Banach–Tarski paradox: A solution to de Groot's Problem

In 1924 Banach and Tarski demonstrated the existence of a paradoxical decomposition of the 3-ball B, i.e., a piecewise isometry from B onto two copies of B. This article answers a question of de Groot from 1958 by showing that there is a paradoxical decomposition of B in which the pieces move continuously while remaining disjoint to yield two copies of B. More generally, we show that if n > 2, any two bounded sets in Rn that are equidecomposable with proper isometries are continuously equidecomposable in this sense.

SOLUTIONS TO CONGRUENCES USING SETS WITH THE PROPERTY OF BAIRE

Hausdorff's paradoxical decomposition of a sphere with countably many points removed (the main precursor of the Banach–Tarski paradox) actually produced a partition of this set into three pieces A,B,C such that A is congruent to B (i.e. there is an isometry of the set which sends A to B), B is congruent to C, and A is congruent to B ∪ C. While refining the Banach–Tarski paradox, R. Robinson characterized the systems of congruences like this which could be realized by partitions of the sphere with rotations witnessing the congruences: the only nontrivial restriction is that the system should not require any set to be congruent to its complement. Later, J. F. Adams showed that this restriction can be removed if one allows arbitrary isometries of the sphere to witness the congruences. The purpose of this paper is to characterize those systems of congruences which can be satisfied by partitions of the sphere or related spaces into sets with the property of Baire. A paper of Dougherty and Foreman gives a proof that the Banach–Tarski paradox can be achieved using such sets, and gives versions of this result using open sets and related results about partitions of spaces into congruent sets. The same method is used here; it turns out that only one additional restriction on a system of congruences is needed to make it solvable using subsets of the sphere with the property of Baire (or solvable with open sets if one allows meager exceptions to the congruences and the covering of the space) with free rotations witnessing the congruences. Actually, the result applies to any complete metric space acted on in a sufficiently free way by a free group of homeomorphisms. We also characterize the systems solvable on the sphere using sets with the property of Baire but allowing all isometries.