 However, in a 1929 paper, von Neumann proved that paradoxical decompositions could use a group of transformations that include as a subgroup a free group with two generators.The group of area-preserving transformations contains such subgroups, and this opens the possibility of performing paradoxical decompositions using these subgroups.The class of groups isolated by von Neumann in his work on Banach–Tarski decompositions subsequently was very important for many areas of mathematics, including von Neumann's own later work in measure theory (see below).