Measurable equidecompositions for group actions with an expansion property (Q2098198)

Summary: Given an action of a group \(\Gamma\) on a measure space \(\Omega\), we provide a sufficient criterion under which two sets \(A , B \subset{\Omega}\) are \textit{measurably equidecomposable}, i.e., \(A\) can be partitioned into finitely many measurable pieces which can be rearranged using some elements of \(\Gamma\) to form a partition of \(B\). In particular, we prove that every bounded measurable subset of \(\mathbb{R}^n\), \(n \geq 3\), with non-empty interior is measurably equidecomposable to a ball via isometries. The analogous result also holds for some other spaces, such as the sphere or the hyperbolic space of dimension \(n \geq 2\).