Chapter 6 - Free Groups of Large Rank: Getting a Continuum of Spheres from One

Summary

The Banach-Tarski Paradox shows how to obtain two spheres, or balls, from one, and it is clear how to get any finite number of balls: just duplicate repeatedly, lifting the subsets of the new balls back to the original. After all, the joy in owning a duplicating machine is being able to use it more than once. Alternatively, one need only consider the weak system of congruences: A_2j ≃ ∪ {A_i: 1 ≤ i ≤ 2n, i ≠ 2j - 1}, j = 1, …, n. By the remarks following 4.12, there is a partition of the sphere into sets A_i that satisfy this system with respect to rotations, and therefore A_2j ∪ A_2j-1 ∼ S². Need we stop at just finitely many copies? What about infinitely many, even uncountably many? Using the existence of infinitely many independent rotations (σⁱ τσ^-i, i = 0, 1, 2, … where σ, τ are independent), it is not hard to see how the results of Chapter 4 on systems of congruences can be made to yield the solvability of any countably infinite weak system by a partition of S². Hence S² can be partitioned into countably many sets, each of which is SO₃-equidecomposable (using just two pieces) with S². But even stronger transfinite duplications are possible. One can get a continuum of spheres from one: the sphere can be partitioned into sets B_α, as many sets as there are points on the sphere (i.e., 2^ℵo), such that each B_α is SO₃-equidecomposable with the sphere.