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 $A,B\subseteq \Omega$ are measurably equidecomposable, i.e., $A$ can be partitioned into finitely many measurable pieces which can be rearranged using the elements of $\Gamma$ to form a partition of $B$. In particular, we prove that every bounded measurable subset of $R^n$, $n\ge 3$, 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 $n\ge 2$.

55 pages, 2 figures

References in corpus (1)

• Baire measurable paradoxical decompositions via matchings

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