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
Measurable equidecompositions for group actions with an expansion property cover
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Abstract

Given an action of a group Γ\Gamma on a measure space Ω\Omega, we provide a sufficient criterion under which two sets A,BΩA, B\subset \Omega are measurably equidecomposable\textit{measurably equidecomposable}, i.e., AA can be partitioned into finitely many measurable pieces which can be rearranged using some elements of Γ\Gamma to form a partition of BB. In particular, we prove that every bounded measurable subset of Rn\mathbb{R}^n, n3n\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 n2n\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