Chapter 5 - Higher Dimensions and Non-Euclidean Spaces

Summary

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 G_n-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).