Equidecomposability of Jordan domains under groups of isometries

• Autorzy: M. Laczkovich
• Języki publikacji: EN

Abstrakty

EN

Let $G_d$ denote the isometry group of $ℝ^d$. We prove that if G is a paradoxical subgroup of $G_d$ then there exist G-equidecomposable Jordan domains with piecewise smooth boundaries and having different volumes. On the other hand, we construct a system $ℱ_d$ of Jordan domains with differentiable boundaries and of the same volume such that $ℱ_d$ has the cardinality of the continuum, and for every amenable subgroup G of $G_d$, the elements of $ℱ_d$ are not G-equidecomposable; moreover, their interiors are not G-equidecomposable as geometric bodies. As a corollary, we obtain Jordan domains A,B ⊂ ℝ² with differentiable boundaries and of the same area such that A and B are not equidecomposable, and int A and int B are not equidecomposable as geometric bodies. This gives a partial solution to a problem of Jan Mycielski.

Kategorie tematyczne

• 43A07: Means on groups, semigroups, etc.; amenable groups
• 05A18: Partitions of sets
• 28A99: None of the above, but in this section
• 52A20: Convex sets in n dimensions (including convex hypersurfaces)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2003
• Tom: 177
• Numer: 2
• Strony: 151-173

Daty

wydano

2003

Twórcy

autor

M. Laczkovich

• Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117 Budapest, Hungary
• Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, England

Identyfikatory

DOI

10.4064/fm177-2-4