Groups of piecewise projective homeomorphisms

I 1924, Banach and Tarski (1) accomplished a rather paradoxical feat. They proved that a solid ball can be decomposed into five pieces, which are then moved around and reassembled in such a way as to obtain two balls identical to the original one (1). This wellnigh miraculous duplication was based on Hausdorff’s (2) 1914 work. In his 1929 study of the Hausdorff–Banach–Tarski paradox, von Neumann (3) introduced the concept of amenable groups. Tarski (4, 5) readily proved that amenability is the only obstruction to paradoxical decompositions. However, the known paradoxes relied more prosaically on the existence of nonabelian free subgroups. Therefore, the main open problem in the subject remained for half a century to find nonamenable groups without free subgroups. Von Neumann’s (3) name was apparently attached to it by Day in the 1950s. The problem was finally solved around 1980: Ol′shanskiĭ (6–8) proved the nonamenability of the Tarski monsters that he had constructed, and Adyan (9, 10) showed that his work on Burnside groups yields nonamenability. Finitely presented examples were constructed another 20 y later by Ol′shanskiĭ–Sapir (11). There are several more recent counterexamples (12–14). Given any subring A < R, we shall define a group G(A) and a subgroup H(A) < G(A) of piecewise projective transformations. Those groups will provide concrete, uncomplicated examples with many additional properties. Perhaps ironically, our short proof of nonamenability ultimately relies on basic free groups of matrices, as in Hausdorff’s (2) 1914 paradox, although the Tits (15) alternative shows that the examples cannot be linear themselves.