Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Fundamenta Mathematicae / All issues

Abstract

Let $G_d$ denote the isometry group of ${\mathbb R}^d.$ 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 ${\cal F}_d$ of Jordan domains with differentiable boundaries and of the same volume such that ${\cal F}_d$ has the cardinality of the continuum, and for every amenable subgroup $G$ of $G_d,$ the elements of ${\cal F}_d$ are not $G$-equidecomposable; moreover, their interiors are not $G$-equidecomposable as geometric bodies. As a corollary, we obtain Jordan domains $A,B\subset {\mathbb R}^2$ with differentiable boundaries and of the same area such that $A$ and $B$ are not equidecomposable, and $\mathop {\rm int} A$ and $\mathop {\rm int} B$ are not equidecomposable as geometric bodies. This gives a partial solution to a problem of Jan Mycielski.