# Measurable equidecompositions for group actions with an expansion property

### Łukasz Grabowski

Lancaster University, UK### András Máthé

University of Warwick, Coventry, UK### Oleg Pikhurko

University of Warwick, Coventry, UK

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

## Cite this article

Łukasz Grabowski, András Máthé, Oleg Pikhurko, Measurable equidecompositions for group actions with an expansion property. J. Eur. Math. Soc. 24 (2022), no. 12, pp. 4277–4326

DOI 10.4171/JEMS/1189