On Gardner's conjecture

Abstract

Gardner conjectured that if two bounded measurable sets $A,B \subset \mathbb{R}^n$ are equidecomposable by a set of isometries $\Gamma$ generating an amenable group then $A$ and $B$ admit a measurable equidecomposition by all isometries. Cie\'sla and Sabok asked if there is a measurable equidecomposition using isometries only in the group generated by $\Gamma$. We answer this question negatively.