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Fundamenta Mathematicae

2003 | 177 | 2 | 151-173

Equidecomposability of Jordan domains under groups of isometries

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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.

151-173

wydano
2003

Twórcy

autor
• 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