In this question, I recently asked if there were free subgroups of rank 3 or higher of the group of rotations in $\mathbb{R}^3$. From the answers, it follows that any free subgroup of rank 2 admits subgroups of arbitrary countable rank.
My question now is whether this can be used to extend the Banach-Tarski Paradox to show that the sphere cannot only be duplicated (leveraging the subgroup of rotations of rank 2), but may be done in a way to produce $n$ copies using a finite number of disjoint subsets of the original unit sphere (leveraging a subgroup of rotations of rank $n$). All I have seen in this vein is re-applying the original statement $n$ times in order to create $n$ spheres, but it seems they could be created all at once using a free subgroup of rank $n$, correct?