19.59 (2018)

Does the group of isometries of $\mathbb{Q}^3$ have a free non-abelian subgroup such that every non-trivial element acts without nontrivial fixed points in $\mathbb{Q}^3$?

The answer is yes if the field of rational numbers $\mathbb{Q}$ is replaced by the field $\mathbb{R}$ of real numbers (see G. Tomkowicz, S. Wagon, The Banach–Tarski Paradox, Cambridge Univ. Press, 2016.)