Von Neumann paradox in the Euclidean plane

Statement

In the Euclidean plane, two figures that are equidecomposable with respect to the group of Euclidean motions are necessarily of the same area, and therefore, a paradoxical decomposition of a square or disk of Banach–Tarski type that uses only Euclidean congruences is impossible. Von Neumann gave a conceptual explanation of the distinction between the planar and higher-dimensional cases: unlike the group SO(3) of rotations in three dimensions, the group E(2) of Euclidean motions of the plane is solvable, which implies the existence of a finitely additive measure on E(2) and R² that is invariant under translations and rotations, and rules out paradoxical decompositions of non-negligible sets. Von Neumann then asked: can such a paradoxical decomposition be constructed if one allows a larger group of equivalences?

It is clear that if one permits similarities, any two squares in the plane become equivalent even without further subdivision. This motivates restricting one's attention to the group SA₂ of area-preserving affine transformations. Since the area is preserved, any paradoxical decomposition of a square with respect to this group would be counterintuitive for the same reasons as the Banach–Tarski decomposition of a ball. In fact, the group SA₂ contains as a subgroup the special linear group SL(2,R), which in turn contains the free group F₂ with two generators as a subgroup. This makes it plausible that the proof of Banach–Tarski paradox can be imitated in the plane. The main difficulty here is that the unit square is not invariant under the action of the linear group SL(2, R), so one cannot simply transfer a paradoxical decomposition from the group to the square, as in the third step of the above proof of the Banach–Tarski paradox. Moreover, the fixed points of the group present difficulties (for example, the origin is fixed under all linear transformations). This is why von Neumann used the larger group SA₂ including the translations, and constructed a paradoxical decomposition of the unit square with respect to the enlarged group (in 1929). Applying the Banach–Tarski method, the paradox for the square can be strengthened as follows:

Any two bounded subsets of the Euclidean plane with non-empty interiors are equidecomposable with respect to the area-preserving affine maps.

As von Neumann notes:

Infolgedessen gibt es bereits in der Ebene kein nichtnegatives additives Maß (wo das Einheitsquadrat das Maß 1 hat), das gegenüber allen Abbildungen von A₂ invariant wäre.

(In accordance with this, already in the plane there is no non-negative additive measure (for which the unit square has a measure of 1), which is invariant with respect to all transformations belonging to A₂ [the group of area-preserving affine transformations].)

To elaborate, whether a finitely additive measure (that is preserved under certain transformations) exists depends on what transformations are allowed. The Banach measure of sets in the plane, which is preserved by translations and rotations, is not preserved by non-isometric transformations even when they do preserve the area of polygons. The points of the plane (other than the origin) can be divided into two dense sets that may be called A and B. If the A points of a given polygon are transformed by a certain area-preserving transformation and the B points by another, both sets can become subsets of the A points in two new polygons. The new polygons have the same area as the old polygon, but the two transformed sets cannot have the same measure as before (since they contain only part of the A points), and therefore no measure "works".

The class of groups von Neumann isolated in the course of studying the Banach–Tarski phenomenon turned out to be very important in many areas of mathematics: these are amenable groups, or groups with an invariant mean, and include all finite and all solvable groups. Generally speaking, paradoxical decompositions arise when the group used for equivalences in the definition of equidecomposability is not amenable.

• 2000: Von Neumann's paper left open the possibility of a paradoxical decomposition of the interior of the unit square with respect to the linear group SL(2,R) (Wagon, Question 7.4). In 2000, Miklós Laczkovich proved that such a decomposition exists. More precisely, let A be the family of all bounded subsets of the plane with non-empty interior and at a positive distance from the origin, and B the family of all planar sets with the property that a union of finitely many translates under some elements of SL(2, R) contains a punctured neighborhood of the origin. Then all sets in the family A are SL(2, R)-equidecomposable, and likewise for the sets in B. It follows that both families consist of paradoxical sets.
• 2003: It had long been known that the full plane was paradoxical with respect to SA₂, and that the minimal number of pieces would equal four provided that there exists a locally commutative free subgroup of SA₂. In 2003 Kenzi Satô constructed such a subgroup, confirming that four pieces suffice.
• 2011: Laczkovich's paper left open the possibility that there exists a free group F of piecewise linear transformations acting on the punctured disk D \ {(0,0)} without fixed points. Grzegorz Tomkowicz constructed such a group, showing that the system of congruences A ≈ B ≈ C ≈ B U C can be realized by means of F and D \ {(0,0)}.
• 2017: It has long been known that there exists in the hyperbolic plane H² a set E that is a third, a fourth and ... and a $2^{\aleph_0}$-th part of H². The requirement was satisfied by orientation-preserving isometries of H². Analogous results were obtained by John Frank Adams and Jan Mycielski, who showed that the unit sphere S² contains a set E that is a half, a third, a fourth and ... and a $2^{\aleph_0}$-th part of S². Tomkowicz showed that Adams and Mycielski construction can be generalized to obtain a set E of H² with the same properties as in S².
• 2017: Von Neumann's paradox concerns the Euclidean plane, but there are other classical spaces where the paradoxes are possible. For example, one can ask if there is a Banach–Tarski paradox in the hyperbolic plane H². This was shown by Mycielski and Tomkowicz. Tomkowicz also proved that most of the classical paradoxes are an easy consequence of a graph theoretical result and the fact that the groups in question are rich enough.
• 2018: In 1984, Mycielski and Stan Wagon constructed a paradoxical decomposition of the hyperbolic plane H² that uses Borel sets. The paradox depends on the existence of a properly discontinuous subgroup of the group of isometries of H². A similar paradox was obtained in 2018 by Tomkowicz, who constructed a free properly discontinuous subgroup G of the affine group SA(3,Z). The existence of such a group implies the existence of a subset E of Z³ such that for any finite F of Z³ there exists an element $g$ of $G$ such that $g(E) = E \triangle F$, where $E\,\triangle\,F$ denotes the symmetric difference of $E$ and $F$.
• 2019: Banach–Tarski paradox uses finitely many pieces in the duplication. In the case of countably many pieces, any two sets with non-empty interiors are equidecomposable using translations. But allowing only Lebesgue measurable pieces one obtains: If A and B are subsets of Rⁿ with non-empty interiors, then they have equal Lebesgue measures if and only if they are countably equidecomposable using Lebesgue measurable pieces. Mycielski and Tomkowicz extended this result to finite dimensional Lie groups and second countable locally compact topological groups that are totally disconnected or have countably many connected components.
• 2024: Tomkowicz and Robert Samuel Simon introduced a coloring rule of points in a Cantor space that links paradoxical decompositions with optimization. This allows one to find an application of paradoxical decompositions in economics.
• 2024: Tomkowicz proved that in the case of non-supramenable connected Lie groups $G$ acting continuously and transitively on a metric space, bounded $G$ paradoxical sets are generic.