The Banach-Tarski theorem on polygons, and the cancellation law

Abstract:

The Banach-Tarski theorem on polygons in ${{\mathbf {R}}^2}$ implies that two polygons are equidecomposable if and only if they are equidissectable. The possibility of strengthening this result in various ways is investigated. We show that if two polytopes in ${{\mathbf {R}}^d}$ are equidecomposable under a finite set of isometries which generates a discrete group, then they are equidissectable using the same isometries. We then give a simple example in ${\mathbf {R}}$ showing that this is not true for arbitrary finite sets of isometries. A modification of this example is used to answer a question of S. Wagon concerning the cancellation law.