TITLE: Can the Banach-Tarski Paradox be extended to an arbitrary number of duplications?
QUESTION [9 upvotes]: 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?

REPLY [8 votes]: This question is answered in the affirmative in Chapter 6 of Stan Wagon's book on the Banach-Tarski Paradox, "Free Groups of Large Rank: Getting a Continuum of Spheres from One."
Using the free subgroups of rank $n$ (of a free group of rank 2), one can extend BT to $n$ copies at once.
Furthermore, as stated in the comments of the above-referenced question, one can use the fact that a free group of rank 2 has a subgroup of countably infinite rank to produce a countably infinite number of copies of the original sphere.
Moreover, it is possible to produce copies of the ball with cardinality of the continuum, though I do not understand the development presented in the text well enough to summarize here.