Groups of piecewise projective homeomorphisms Nicolas Monod1 Section de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland Edited by Gregory A. Margulis, Yale University, New Haven, CT, and approved February 5, 2013 (received for review October 22, 2012) The group of piecewise projective homeomorphisms of the line The main result of this article is the following, for which we provides straightforward torsion-free counterexamples to the so- introduce a method for proving amenability. called von Neumann conjecture. The examples are so simple that many additional properties can be established. Theorem 1. The group H(A) is nonamenable if A ≠ Z. The next result is a sequacious generalization of the corre- free groups | paradoxical decomposition | von Neumann problem sponding theorem of Brin–Squier (16) about piecewise affine transformations, and we claim no originality. n 1924, Banach and Tarski (1) accomplished a rather para- Idoxical feat. They proved that a solid ball can be decomposed Theorem 2. The group H does not contain any nonabelian free sub- fi group. Thus, H(A) inherits this property for any subring A < R. into ve pieces, which are then moved around and reassembled = R in such a way as to obtain two balls identical to the original one Thus, already H H( ) itself is a counterexample to the von Neumann conjecture. Writing H(A) as the directed union of its (1). This wellnigh miraculous duplication was based on Haus- fi dorff’s (2) 1914 work. nitely generated subgroups, we deduce Corollary 3. In his 1929 study of the Hausdorff–Banach–Tarski paradox, Corollary 3. For A ≠ Z, the groups H(A) contain finitely generated von Neumann (3) introduced the concept of amenable groups. subgroups that are simultaneously nonamenable and without non- Tarski (4, 5) readily proved that amenability is the only ob- abelian free subgroups. struction to paradoxical decompositions. However, the known paradoxes relied more prosaically on the existence of nonabelian Additional Properties. The groups H(A) seem to enjoy a number free subgroups. Therefore, the main open problem in the subject of additional interesting properties, some of which are weaker remained for half a century to find nonamenable groups without forms of amenability. In the last section, we shall prove the fol- free subgroups. Von Neumann’s (3) name was apparently at- lowing five propositions (and recall the terminology). Here, A < R tached to it by Day in the 1950s. The problem was finally solved is an arbitrary subring. around 1980: Ol′shanskiĭ (6–8) proved the nonamenability of the Tarski monsters that he had constructed, and Adyan (9, 10) Proposition 4. All L2-Betti numbers of H(A) and G(A) vanish. showed that his work on Burnside groups yields nonamenability. Finitely presented examples were constructed another 20 y later Proposition 5. The group H(A) is inner amenable. by Ol′shanskiĭ–Sapir (11). There are several more recent coun- terexamples (12–14). Proposition 6. The group H is biorderable, and hence, so are all of its Given any subring A < R,weshalldefine a group G(A)anda subgroups. It follows that there is no nontrivial homomorphism subgroup H(A) < G(A) of piecewise projective transformations. from any Kazhdan group to H. Those groups will provide concrete, uncomplicated examples with many additional properties. Perhaps ironically, our short Proposition 7. Let E ⊆ P1 be any subset. Then, the subgroup of proof of nonamenability ultimately relies on basic free groups H(A), which fixes E pointwise, is coamenable in H(A) unless E is of matrices, as in Hausdorff’s (2) 1914 paradox, although the dense (in which case, the subgroup is trivial). Tits (15) alternative shows that the examples cannot be linear themselves. Proposition 8. If H(A) acts by isometries on any proper CAT(0) space, then either it fixesapointatinfinity or it preserves a Euclidean subspace. Construction One can also check that H(A) satisfies no group law and has vanishing properties in bounded cohomology (see below). I saw the pale student of unhallowed arts kneeling beside the thing he Nonamenability had put together. An obvious difference between the actions of PSL2(A) and H(A) on P1 is that the latter group fixes ∞, whereas the former does Mary Shelley, introduction to the 1831 edition of Frankenstein not. The next proposition shows that this difference is the only Consider the natural action of the group PSL2(R)onthe one as far as the orbit structure is concerned. projective line P1 = P1(R). We endow P1 with its R-topology, 1 making it a topological circle. We denote by G the group of all Proposition 9. Let A < R be any subring, and let p ∈ P \{∞}. Then, 1 homeomorphisms of P that are piecewise in PSL2(R), each piece 1 · ⊆ ∞ ∪ · : being an interval of P with finitely many pieces. We let H < G PSL2ðAÞ p f g HðAÞ p be the subgroup fixing the point ∞∈P1 corresponding to the first basis vector of R2. Thus, H is left-orderable, because it acts Thus, the equivalence relations induced by the actions of PSL2(A) P1 P1 ∞ faithfully on the topological line P1\{∞}, preserving orientations. and H(A) on coincide when restricted to \{ }. It follows in particular that H is torsion-free. 1 Given a subring A < R, we denote by PA ⊆ P the collection of Author contributions: N.M. designed research, performed research, and wrote the paper. all fixed points of all hyperbolic elements of PSL2(A). This set is The author declares no conflict of interest. PSL2(A)-invariant and countable if A is so. We define G(A)tobe the subgroup of G given by all elements that are piecewise in This article is a PNAS Direct Submission. PSL2(A) with all interval endpoints in PA. We write H(A) = G(A) Freely available online through the PNAS open access option. ∩ H, which is the stabilizer of ∞ in G(A). 1E-mail: nicolas.monod@epfl.ch. 4524–4527 | PNAS | March 19, 2013 | vol. 110 | no. 12 www.pnas.org/cgi/doi/10.1073/pnas.1218426110 Downloaded by guest on October 1, 2021 Proof. We need to show that, given g ∈ PSL2(A)withgp ≠∞, nonamenable (see also Remark 10 and Remark 11). Equiva- ∈ = 1 there is an element h H(A), such that hp gp. We assume lently, the Γ-actiononPSL2(R) is nonamenable. Viewing P as ∞≠∞ = g , because otherwise, h g will do. Equivalently, we a homogeneous space of PSL2(R), it follows that the Γ-action on need an element q ∈ G(A) fixing gp andsuchthatq∞≠g∞, P1 is nonamenable. Indeed, amenability is preserved under −1 writing h ≠ q g.Itsuffices to find a hyperbolic element q0 ∈ extensions (ref. 21, Corollary C or ref. 23, 2.4). This action is a. 1 PSL2(A)withq0∞ = g∞ and fixed points ξ± ∈ P that separate e.-free, because any nontrivial element has, at most, two fixed gp from both ∞ and g∞ (Fig. 1). Indeed, we can then define q points. Thus, the relation induced by Γ on P1 is nonamenable. 1 1 to be the identity on the component of P \{ξ±} containing gp Restricting to P \{∞}, we deduce from Proposition 9 that the fi and de ne qto coincide with q0 on the other component. relation induced by the H(A) action is also nonamenable. ab Let be a matrix representative of g; thus, a, b, c, d, ∈ A, [Amenability is preserved under restriction (ref. 18, 9.3), but here, cd {∞} is a null-set anyway.] Thus, H(A) is a nonamenable group.□ − = ∞≠∞ ≠ and ad bc 1. The assumption g implies c 0, and ab+ ra Remark 10. We recall from ref.22that the nonamenability of the thus, we can assume c > 0. Let q0 be given by with cd+ rc Γ-relation on PSL2(R) is a general consequence of the existence of Γ r ∈ A to be determined later; thus, q0∞≠g∞. This matrix is a nondiscrete nonabelian free subgroup of . Thus, the main point hyperbolic as soon as jrj is large enough to ensure that the trace of our appeal to ref.22is the existence of this nondiscrete free τ = a + d + rc is larger than 2 in absolute value. We only need to subgroup, but this existence is much easier to prove directly in the show that a suitable choice of r will ensure the above condition present case of Γ = PSL2(A) than for general nondiscrete 1 on ξ±. Notice that ∞ and g∞ lie in the same component of P \{ξ±}, nonsoluble Γ. because q0 preserves these components and sends ∞ to g∞.In fi conclusion, it suf ces to prove the following two claims: (i)as Remark 11. Here is a direct argumentpffiffiffi avoiding all of the above ref- jrj → ∞,theset{ξ±}convergesto{∞, g∞}; and (ii) changing erences in the examples of A = Z½ 2 or A = Z[1/ℓ], where ℓ is the sign of r (when jrj is large) will change the component of prime. We show directly that the Γ-action on P1 is not amenable. P1 ∞ ∞ ξ \{ , g } in which ± lie (we need it to be the component of gp). We consider Γ as a lattice in L: = PSL2(R) × PSL2(R) in the first The claims can be proved by elementary dynamical considera- case and L: = PSL2(R) × PSL2(Qℓ) in the second case, both times tions; we shall instead verify them explicitly. in such a way that the Γ-action on P1 extends to the L-action x ± 1 The fixed points ξ± are represented by the eigenvectors , factoring through the first factor. If the Γ-action on P were ame- pffiffiffiffiffiffiffiffiffiffiffiffi c where x± = λ± − d − rc and λ ± = ðτ ± τ2 − 4Þ=2 are the eigen- nable, so would be the L-action (by coamenability of the lattice).