Measurable equidecompositions for group actions with an expansion property

TL;DR

This paper establishes criteria for measurable equidecomposability of sets under group actions with an expansion property, including a proof that bounded sets with interior in higher dimensions are equidecomposable to a ball.

Contribution

It introduces a sufficient condition for measurable equidecomposability under group actions with expansion properties, extending classical results to new spaces and settings.

Findings

Sets with non-empty interior in R^n are equidecomposable to a ball via isometries.

The criterion applies to various spaces like spheres and hyperbolic spaces.

Every bounded measurable set with interior in R^n (n≥3) is equidecomposable to a ball.

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 $n\ge 2$.