I don't know if this is even a legitimate question. The Banach-Tarski Paradox has been answered in terms of Amenable groups. Consider G-sets (pseudo-groups) as actions of groups on sets. Now consider the Poincare Group (general relativity) action on 4-space (Pseudo-Riemann manifold which is a Lie group.) Now compare this action to the action of a Poincare group on an exotic R4 (also a Lie group). Does there exist a paradoxical decomposition for the pseudo-groups in question? In other words, can one instantiate the Banach-Tarski paradox for the Poincare action on an exotic R4s? For that matter, does there exist paradoxical decompositions for the Poincare action on a Pseudo-Riemann manifold as well?
Like I said, I don't even know if even makes any sense to pose the question this way. Certainly, one must be more restrictive with regard to considerations of compactness and connectedness with regard to the Poincare group to begin with.
Perhaps, some clarity in terms of issues of surjectivity and injectivity of homomorphisms from the Poincare group to the group of all automorphisms of the Lie group in question(i.e. pseudo-Riemann manifold vs exotic R4) would be helpful in fiber bundle language or co-homological descriptions. I'm assuming here that the Lie groups are not only manifolds but smooth manifolds (Hilbert's 5th problem answered in the affirmative.)
I guess what I'm driving at is whether the whole issue of exotic smoothness can have anything to do with general relativity. Certainly 4 manifolds support a large number of exotic forms of smoothness because there is no generalized Poincare Conjecture that applies to diffeomorphisms of R4s. The simple fact is that Einstein's notion of general covariance may apply to every dimension of manifold except the very one (4 dimensions) that we need.
