A class of free rotation groups
The following theorem was announced by the author in 1958: if \(\cos \varphi \in \mathbb{Q}\) then the subgroup of \(\text{SO}_ 3(\mathbb{R})\) generated by two rotations about the angle \(\varphi\), with rotation axes perpendicular to each other is free if \(\cos \varphi \neq 0\), \(\pm{1\over 2},\pm 1\). The present paper contains the proof of this theorem. As a consequence two-generated free subgroups of \(\text{SO}_ 3 (\mathbb{Q})\) are constructed. Such a group is of interest, as it can be employed to construct paradoxical decompositions of the rational sphere \(S^ 2 \cap \mathbb{Q}^ 3\) without assuming the Axiom of Choice [\textit{S. Wagon}, The Banach-Tarski paradox (Encycl. Math. Appl. 24, Cambridge 1985; Zbl 0569.43001)]. Another consequence of the above theorem is also noteworthy. Let \(\cos(\pi \mathbb{Q}) = \{\cos (\pi x) : x \in \mathbb{Q}\}\) and let \(\sin (\pi \mathbb{Q})\) and \(\tan (\pi \mathbb{Q})\) be defined similarly. Then 1. \(\mathbb{Q} \cap \cos (\pi \mathbb{Q}) = \mathbb{Q} \cap \sin (\pi \mathbb{Q}) = \{0, {1\over 2}, \pm 1\}\). 2. \(\mathbb{Q} \cap \tan (\pi \mathbb{Q}) = \{0, \pm 1\}\). The last result was obtained independently by P. Walker.
scientific article; zbMATH DE number 639726
scientific article