Measurable equidecompositions for group actions with an expansion property

Abstract

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 $\text{measurably equidecomposable}$, 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 ${\mathbb{R}}^n$, $n\ge 3$, 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 $n\ge 2$.