Free product \(\mathbb{Z}_3*\mathbb{Z}_3\) of rotations with rational entries (Q1973927)

The existence of free products and free groups in a rotation group is the key step in establishing the Banach-Tarski paradox [see \textit{S. Wagon}, The Banach-Tarski paradox, Cambridge University Press, Cambridge (1985; Zbl 0569.43001)]. The authors develop an approach to construct the free product \(\mathbb{Z}_3*\mathbb{Z}_3\) as a subgroup of the 3-dimensional special orthogonal group over the rational numbers, avoiding transcendental numbers and irrational numbers.