1 Free Group of Rotations
1.1 Selection of Rotations and Decomposition of the Majority of the Sphere
We specifically choose two rotations of the sphere, with which we will construct the decomposition in the theorem.
Our rotation matrices are:
It holds that \(\det A \neq 0\) and \(\det B \neq 0\), and thus \(A\) and \(B\) are invertible.
This follows by direct calculation.
\(A\) and \(B\) therefore generate a subgroup of the group of invertible \(3 \times 3\) matrices.
Let \(G\) denote the subgroup generated by \(A\) and \(B\).
The adjugate of a \(3 \times 3\) matrix can be represented in a specific form...
If \(\rho : \mathbb {R}^3 \rightarrow \mathbb {R}^3\) is an expression in \(G\) of length \(n\) in reduced form, then \(\rho (0,1,0)\) is of the following form, where \(a\), \(b\), and \(c\) are integers: \(\rho (0,1,0) = \frac{1}{3^n}(a\sqrt{2}, b, c\sqrt{2})\).
This assertion follows from the generator matrices and by explicitly multiplying a reduced word by \((0,1,0)\).
With this, we can show that this subgroup of rotations is a free group with two generators.
A free group \(G\) is a group in which two words on a specific generating set are different, except when their equality follows from the group axioms.
The subgroup \(G\) generated by our specific rotations from Definition 1 is a free group.
This follows directly.