Je travaille sur la caractérisation des groupes qui permettent des décompositions paradoxales dans le plan euclidien. Je comprends que les groupes admissibles sont ceux qui possèdent une moyenne invariante, mais j'ai du mal à établir formellement pourquoi l'absence d'une telle moyenne conduit nécessairement à des décompositions paradoxales. Quel est le lien conceptuel exact entre l'admissibilité d'un groupe et l'existence de décompositions paradoxales de carrés dans le plan ?
Von Neumann paradox in the Euclidean plane

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. A conceptual explanation of the distinction between the planar and higher-dimensional cases was given by John von Neumann: 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 R which is invariant under translations and rotations, and rules out paradoxical decompositions of non-negligible sets. Von Neumann then posed the following question: 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 SA 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 SA contains as a subgroup the special linear group SL (2, R), which in its turn contains the free group F 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 lies in the fact that the unit square is not invariant under the action of the linear group SL (2, R), hence 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 SA including the translations, and he 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:

As von Neumann notes:

To explain further, the question of whether a finitely additive measure (that is preserved under certain transformations) exists or not 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 which 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 there is no measure that "works".

The class of groups isolated by von Neumann in the course of study of Banach–Tarski phenomenon turned out to be very important for 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.
Paradoxes

L'hypothèse selon laquelle toute propriété peut être utilisée pour former un ensemble, sans restriction, conduit à des paradoxes. Un exemple courant est le paradoxe de Russell : il n'existe aucun ensemble constitué de « tous les ensembles qui ne se contiennent pas eux-mêmes ». Ainsi, les systèmes cohérents de la théorie naïve des ensembles doivent inclure certaines limitations sur les principes qui peuvent être utilisés pour former des ensembles.
### Query ###
Je travaille sur la caractérisation des groupes qui permettent des décompositions paradoxales dans le plan euclidien. Je comprends que les groupes admissibles sont ceux qui possèdent une moyenne invariante, mais j'ai du mal à établir formellement pourquoi l'absence d'une telle moyenne conduit nécessairement à des décompositions paradoxales. Quel est le lien conceptuel exact entre l'admissibilité d'un groupe et l'existence de décompositions paradoxales de carrés dans le plan ?

### Main Text ###
Von Neumann paradox in the Euclidean plane

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. A conceptual explanation of the distinction between the planar and higher-dimensional cases was given by John von Neumann: 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 R which is invariant under translations and rotations, and rules out paradoxical decompositions of non-negligible sets. Von Neumann then posed the following question: 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 SA 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 SA contains as a subgroup the special linear group SL (2, R), which in its turn contains the free group F 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 lies in the fact that the unit square is not invariant under the action of the linear group SL (2, R), hence 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 SA including the translations, and he 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:

As von Neumann notes:

To explain further, the question of whether a finitely additive measure (that is preserved under certain transformations) exists or not 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 which 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 there is no measure that "works".

The class of groups isolated by von Neumann in the course of study of Banach–Tarski phenomenon turned out to be very important for 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.

### Supplementary Text ###
Paradoxes

L'hypothèse selon laquelle toute propriété peut être utilisée pour former un ensemble, sans restriction, conduit à des paradoxes. Un exemple courant est le paradoxe de Russell : il n'existe aucun ensemble constitué de « tous les ensembles qui ne se contiennent pas eux-mêmes ». Ainsi, les systèmes cohérents de la théorie naïve des ensembles doivent inclure certaines limitations sur les principes qui peuvent être utilisés pour former des ensembles.

### Draft ###

### 1. Query Decomposition

"Je travaille sur la caractérisation des groupes qui permettent des décompositions paradoxales..."

Translation: "I'm working on characterizing groups that allow paradoxical decompositions in Euclidean plane."

Key terms:
- décompositions paradoxales = paradoxical decompositions  
- groupes admissibles = admissible groups
- moyenne invariante = invariant mean

? User conflates "admissible groups" with groups having invariant mean. Need clarification.

### 2. Mathematical Context Mapping

● Banach-Tarski paradox (1924): Disjoint pieces → full 3D ball
○ Plane version: More constrained due to area preservation
● von Neumann (1927): Solvable groups have invariant measures

→ Core tension: How can area-preserving transformations preserve measure?

### 3. The Constraint Hierarchy

**Euclidean motions E(2):**
- Translations: T(x) = x + v
- Rotations: R(x) = Ax (orthogonal A)  
- Area-preserving ✓
- Invariant mean exists ✓  
- No paradoxical squares possible ✓

**Affine area-preserving SA:**  
- Includes translations + rotations + uniform scaling
- Still area-preserving ✓
- But: Non-isometric transformations break measure invariance
- Example: Scaling by λ ≠ 1 creates different "measure density"

### 4. The Measure Theory Gap

⟨H≈0.8⟩ When you scale a square by λ, its area changes by λ². But its "measure content" must stay constant under transformations. This creates the paradox. A set of points that was "dense" in the original square (containing most of the area) can't stay "dense" after scaling - it's now diluted across the larger transformed area.

∴ No finitely additive measure can exist that's both:
- Translation/rotation invariant ✓  
- Area-preserving transformation invariant ✗

### 5. von Neumann's Insight

! Key realization: Solvable groups → invariant mean → measure theory constraints

For E(2): Solvable → amenable → invariant mean exists

For SA: Non-solvable → no invariant mean → paradoxical decompositions possible

### 6. Formal Connection

**Theorem (von Neumann, 1929):**
- E(2) (Euclidean motions) has invariant mean → no paradoxical squares
- SA (area-preserving affine) lacks invariant mean → paradoxical squares exist

**Proof sketch:**
1. Assume paradoxical decomposition of unit square under SA
2. Consider area-preserving transformation T: square → rectangle (different area)  
3. If T preserves measure → contradiction (dense sets have different measure density)
4. ∴ No such paradoxical decomposition exists

### 7. Conceptual Bridge

The exact link:

**Non-amenable groups** (like SA) → **No invariant mean** → **Measure paradoxes possible**

**Amenable groups** (like E(2)) → **Invariant mean exists** → **Paradoxes excluded**

⟨H≈1.2⟩ Think of it as the "conservation law" of geometric measure space. Just as you can't create energy from nothing, you can't create measure from nothing. The transformations stretch space but must preserve the "accounting" of where the measure lives. Non-amenable groups break this accounting system.

### 8. The Larger Context

von Neumann's work → Foundation of ergodic theory
→ Characterization of amenable groups  
→ Deep connection: Which groups "behave well" under averaging?

**Amenable groups include:**
- Finite groups
- Solvable groups  
- Discrete Heisenberg group
- Some lattices in Lie groups

**Non-amenable groups include:**
- Free groups
- Hyperbolic groups
- Some discrete subgroups of SL₂(ℝ)

### 9. Final Synthesis

The conceptual link is **measure-theoretic impossibility**. Non-amenable groups lack the "measurability structure" that would allow area-preserving transformations to preserve meaningful measure. This creates the paradox: you can decompose a square into pieces, but those pieces can't be "reassembled" via area-preserving transformations back to the original square while maintaining consistent measure accounting.

∴ Amenability = measurability = impossibility of true paradoxical decompositions for geometric objects.

### Answer###
Le lien conceptuel que vous recherchez repose sur la distinction fondamentale entre les groupes résolubles et les groupes qui ne possèdent pas de moyenne invariante.

Dans le plan euclidien, les transformations qui préservent l'aire (comme les translations et les rotations) définissent un groupe résoluble appelé E(2). Ce groupe possède une mesure finitairement additive et invariante sous les translations et rotations, ce qui exclut formellement les décompositions paradoxales de carrés ou de disques de type Banach-Tarski.

Cependant, lorsque l'on considère des transformations affines préservant l'aire (groupe SA), la situation change radicalement. Bien que l'aire soit préservée, les transformations non isométriques ne préservent pas la mesure des polygones. Un carré peut être transformé de manière à ce que des ensembles de points denses dans le carré original se retrouvent comme sous-ensembles de nouveaux ensembles de points, mais sans pouvoir conserver leur mesure antérieure.

Cette apparente contradiction s'explique par l'absence de mesure finitairement additive préservée par toutes les transformations du groupe SA. Les points du plan (autres que l'origine) peuvent être divisés en deux ensembles denses A et B. Lorsqu'une transformation affine préservant l'aire applique l'ensemble A selon une certaine transformation et l'ensemble B selon une autre, les nouveaux polygones résultants ont la même aire que l'ancien, mais les ensembles transformés ne peuvent pas avoir la même mesure que précédemment, car ils ne contiennent qu'une partie des points A.

Ainsi, l'absence de moyenne invariante pour le groupe SA, contrairement au groupe E(2), crée les conditions nécessaires pour permettre les décompositions paradoxales du carré. Cette caractérisation des groupes admissibles par leur propriété d'être amenables (ou ayant une moyenne invariante) explique précisément pourquoi les groupes non amenables, comme le groupe SA, supportent des paradoxes de type Banach-Tarski alors que les groupes résolubles comme E(2) ne le font pas.