2022/04/26 by Łukasz Grabowski, András Máthé, Oleg Pikhurko · 1 citation
Mathematics · #Advanced Operator Algebra Research #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematics #Property (philosophy) #Group (periodic table) #Pure mathematics #Group action #Algebra over a field #Combinatorics
paper · pdf · doi:10.4171/jems/1189
openalex publication_date 2022/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Given an action of a group Γ on a measure space Ω , we provide a sufficient criterion under which two sets A, B⊂ Ω are measurably equidecomposable , i.e., A can be partitioned into finitely many measurable pieces which can be rearranged using some elements of Γ to form a partition of B . In particular, we prove that every bounded measurable subset of ℝn , n≥ 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≥ 2 .