Equidecomposability of Jordan domains under groups of isometries

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.

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M. Laczkovich. "Equidecomposability of Jordan domains under groups of isometries." Fundamenta Mathematicae 177.2 (2003): 151-173. <http://eudml.org/doc/282922>.

@article{M2003,
abstract = {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.},
author = {M. Laczkovich},
journal = {Fundamenta Mathematicae},
keywords = {equidecomposable sets; paradox and amenable groups; uniform amenability},
language = {eng},
number = {2},
pages = {151-173},
title = {Equidecomposability of Jordan domains under groups of isometries},
url = {http://eudml.org/doc/282922},
volume = {177},
year = {2003},
}

TY - JOUR
AU - M. Laczkovich
TI - Equidecomposability of Jordan domains under groups of isometries
JO - Fundamenta Mathematicae
PY - 2003
VL - 177
IS - 2
SP - 151
EP - 173
AB - 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.
LA - eng
KW - equidecomposable sets; paradox and amenable groups; uniform amenability
UR - http://eudml.org/doc/282922
ER -

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