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: A2j ≃ ∪ {Ai: 1 ≤ i ≤ 2n, i ≠ 2j - 1}, j = 1, …, n. By the remarks following 4.12, there is a partition of the sphere into sets Ai that satisfy this system with respect to rotations, and therefore A2j ∪ A2j-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 Bα, as many sets as there are points on the sphere (i.e., 2ℵo), such that each Bα is SO3-equidecomposable with the sphere.

In this chapter we shall present some interesting connections between the amenability of a group and the rate of growth of a group, that is, the speed at which new elements appear when one considers longer and longer words using letters from a fixed finite subset of the group. This approach to amenability sheds light on a basic difference between Abelian and solvable groups. Both types of groups are amenable, but their growth properties can be quite different. This will explain why there is a paradoxical subset of the plane (Sierpiński-Mazurkiewicz Paradox, 1.7), but no similar subset of R1.

The study of growth conditions also elucidates the amenability of Abelian groups. One may prove quite simply (without the Axiom of Choice) that an Abelian group is not paradoxical (12.8), but the proof that Abelian groups are amenable (10.4(b)) is more complicated. One can then obtain an alternate proof of 10.4 (b) by applying Tarski's Theorem (9.2).

We shall also discuss some recent work on the cogrowth of a group, a notion that refines the idea of the growth of a group. This leads to a striking and important characterization of amenable groups, a characterization that is central to the proof of Theorem 1.12, which constructs a group in NF\AG.

The notion of amenability of a group is based on the existence of a measure of total measure one. But we are often interested in invariant measures that assign specific subsets measure one.

Since paradoxical decompositions depend on free groups, and since the group of rotations of S2 is contained in higher-dimensional rotation groups, it comes as no surprise that paradoxical decompositions exist for higher-dimensional spaces. This generalization is not completely obvious, though, since the fixed point set of an isometry does expand when the isometry is extended to a higher dimension by fixing additional coordinates. Nevertheless, the basic results from Chapter 3 on the existence of paradoxical decompositions do extend without requiring any new techniques (see Theorem 5.1). For example, we have already seen that the unit ball in Rn is Gn-negligible if n ≥ 3 (see proof of 2.6), and by the theorem of Tarski alluded to just prior to Theorem 2.6, it follows that such balls are paradoxical. But it is useful to see how the decompositions in higher dimensions may be obtained quite directly from the construction on S2, as is done in Theorem 5.1.

The expansion of the fixed point set is a crucial impasse to generalizing the finer analysis of Chapter 4, however. This is because new fixed points completely destroy the local commutativity of a group when it is viewed as acting on a higher-dimensional space. Nonetheless, locally commutative free groups of isometries (and, where possible, free groups without fixed points) do exist; hence there are minimal paradoxical decompositions in all higher dimensions (see 5.5).

In this final chapter, we give a more detailed account of the role played by the Axiom of Choice (AC) in the theory of paradoxical decompositions. Ever since its discovery, the Banach-Tarski Paradox has caused many mathematicians to look critically at the Axiom of Choice. Indeed, as soon as the Hausdorff Paradox was discovered it was challenged because of its use of AC; E. Borel [21, p. 256] objected because the choice set was not explicitly defined. We shall discuss these criticisms in more detail later in this chapter, but first we deal with several technical points that are essential to understanding the connection between AC and the Banach-Tarski Paradox.

Results of modern set theory can be used to show that AC is indeed necessary to obtain the Banach-Tarski Paradox, in the sense that the paradox is not a theorem of ZF alone. Before we can explain why this is so we need to introduce some notation and discuss some technical points of set theory. If T is a collection of sentences in the language of set theory, for example, T = ZF or T = ZF + AC, then Con(T) is the assertion, also a statement of set theory in fact, that T is consistent, that is, that a contradiction cannot be derived from T using the usual methods of proof. We take Con(ZF) as an underlying assumption in all that follows. Gödel proved in 1938 that Con(ZF) implies (and so is equivalent to) Con(ZF + AC); thus AC does not contradict ZF (see [98, 99]).

This book is motivated by the following theorem of Hausdorff, Banach, and Tarski: Given any two bounded sets A and B in three-dimensional space R3, each having nonempty interior, one can partition A into finitely many disjoint parts and rearrange them by rigid motions to form B. This, I believe, is the most surprising result of theoretical mathematics. It shows the imaginary character of the unrestricted idea of a set in R3. It precludes the existence of finitely additive, congruence-invariant measures over all bounded subsets of R3 and it shows the necessity of more restricted constructions such as Lebesgue's measure.

In the 1950s, the years of my mathematical education in Poland, this result was often discussed. J. F. Adams, R. M. Robinson, and W. Sierpiński wrote about it; my Ph. D. thesis was motivated by it. (All this is referenced in this monograph.) Thus it is a great pleasure to introduce you to this book, where this striking theorem and many related results in geometry and measure theory, and the underlying tools of group theory, are presented with care and enthusiasm. The reader will also find some applications of the most recent advances of group theory to measure theory — work of Gromov, Margulis, Rosenblatt, Sullivan, Tits, and others.

But to me the interest of mathematics lies no more in its theorems and theories than in the challenge of its surprising problems.

Certain proofs and theorems involving equidecomposability would be much simplified if we could add sets. For instance, if X could literally be added to X to form 2X, then the fact that X is paradoxical could be stated simply as X = 2X. In fact, this can be done if we expand the group action appropriately so that multiple copies of X can be formed. This new context for discussing equidecomposability will allow us to state and prove theorems that otherwise would be very cumbersome. One of these is a cancellation law for equidecomposability that has several uses, the most important of which is its use in Tarski's theorem (9.2) relating paradoxical decompositions and invariant measures. Another application will be a proof that any two subsets of S2 with nonempty interior are equidecomposable using rotations. This expanded context for equidecomposability will also yield a simpler proof of Theorem 4.5 that a locally commutative action of a free non-Abelian group is paradoxical.

Definition 8.1. Suppose the group G acts on X. Define an enlarged action as follows. Let X* = X × N and let G* = {(g, π): g ∈ G and π is a permutation of N}, and let the group G* act on X* by (g, π)(x, n) = (g(x), π(n)). If A ⊆ X*, then those n ∈ N such that A has at least one element with second coordinate n are called the levels of A.