Equidecomposability of Jordan domains under groups of isometries

M. Laczkovich

M. Laczkovich

Fundamenta Mathematicae, Tome 177 (2003), p. 151-173 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Publié le : 2003-01-01

Zbl 1019.05006

EUDML-ID : urn:eudml:doc:282922

@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm177-2-4,
     author = {M. Laczkovich},
     title = {Equidecomposability of Jordan domains under groups of isometries},
     journal = {Fundamenta Mathematicae},
     volume = {177},
     year = {2003},
     pages = {151-173},
     zbl = {1019.05006},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm177-2-4}
}

M. Laczkovich. Equidecomposability of Jordan domains under groups of isometries. Fundamenta Mathematicae, Tome 177 (2003) pp. 151-173. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm177-2-4/