Paradoxical decompositions of free F 2 $F_2$ -sets and the Hahn-Banach axiom

Abstract

Denoting by $F_2$ the free group over a two-element alphabet, we show in set-theory without the axiom of choice $\mathrm{ZF}$ that the existence of a (2, 2)-paradoxical decomposition of free $F_2$-sets follows from the conjunction of a weakened consequence of the Hahn-Banach axiom and a weakened consequence of the axiom of choice for pairs. The existence in $\mathrm{ZF}$ of a paradoxical decomposition with 4 pieces of the sphere in the 3-dimensional euclidean space follows from the same two statements restricted to the set $R$ of real numbers. Our result is linked to the $(m,n)$-paradoxical decompositions of free $F_2$-sets previously obtained by Pawlikowski ($m=n=3$, cf. [11]) and then by Sato and Shioya ($m=3$ and $n=2$, cf. [13]) with the sole Hahn-Banach axiom.