Paradoxical decompositions of groups and the sets they act upon

Abstract

The Banach-Tarski paradox states that a solid ball in [special characters omitted] can be partitioned in such a way that by simply rotating and translating those pieces, you end up with two exact copies of the original ball you started with. We give a detailed decomposition of nine pieces and nine different isometries that can be used to produce two copies of the unit ball. We also show this paradox is specifically due to the fact that the group of isometries in [special characters omitted] has a free subgroup with two generators. In the second half of this paper, we introduce the property that prevents this paradox from occuring and show the group of isometries in [special characters omitted] and [special characters omitted] possess this property.