Large free groups of isometries and their geometrical uses (Q1057552)

Motivated by some geometric problems related to the Banach-Tarski paradox, the authors investigate the following questions for spheres and Euclidean and hyperbolic spaces: (1) Is there a free group of isometries of rank the continuum no element of which (except the identity) has a fixed point? (2) Is there a free group of isometries of rank the continuum whose action is locally commutative? The authors settle all remaining cases of these two problems in \({\mathbb{R}}^ n\), \(S^ n\), and \({\mathbb{H}}^ n\), using recent work of Borel, Deligne and Sullivan in \(S^ 4\) and \(S^{4k+1}\), and a special argument in the case of \({\mathbb{H}}^ 3\). The geometric applications are discussed. These typically use nonmeasurable sets, but the existence of a rank-two, free, discrete group of isometries in \({\mathbb{H}}^ 2\) is used to construct a paradoxical decomposition of \({\mathbb{H}}^ 2\) using very simple Borel sets. The authors question in {\S} 9 regarding a simple paradoxical decomposition of \({\mathbb{R}}^ 2\setminus \{(0,0)\}\) using transformations in \(SL_ 2({\mathbb{Z}})\) has been resolved by the first author: Let A be the region between the x-axis and the line \(y=x\); let B be the region between the y-axis and the same line; and let C be the remaining set. Then \(A\cup B\equiv A\equiv B\equiv C\).