Hausdorff Paradox/Lemma 1/Proof Outline

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Proof Outline for Hausdorff Paradox: Lemma

Let $G$ be the free product of the groups $G_1 = \set {e_1, \phi}$ and $G_2 = \set {e_2, \psi, \psi^2}$.

Let $U := \mathbb D^3 \subset \R^3$ be a unit ball in real Euclidean space of $3$ dimensions.

Let $\phi$ and $\psi$ be represented by the axes of rotation $a_\phi$ and $a_\psi$ passing through the center of $U$ such that:

$\phi$ is a rotation by $180 \degrees$, that is $\pi$ radians about $a_\phi$
$\psi$ is a rotation by $120 \degrees$, that is $\dfrac {2 \pi} 3$ radians about $a_\psi$

Hence consider $G$ as the group of all rotations generated by $\phi$ and $\psi$.

The identity of $G$ is then the identity mapping $\mathbf I_3$.


Then $a_\phi$ and $a_\psi$ can be determined in such a way that distinct elements of $G$ represent distinct rotations generated by $\phi$ and $\psi$.


Proof Outline

We determine the angle $\theta$ between $a_\phi$ and $a_\psi$ such that no element of $G$ other than its identity $\mathbf I_3$ represents the identity rotation.

Let us consider a typical element $\alpha$ of $G$:

$\alpha = \phi \circ \psi^{\pm 1} \circ \cdots \circ \phi \circ \psi^{\pm 1}$

Using the properties of orthogonal transformations and elementary trigonometry, we seek to prove that the equation:

$\alpha = \mathbf I_3$

has only finitely many solutions.

Consequently, except for a countable set of values, we may select any angle $\theta$ that satisfies the requirements.


Sources

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