Groups of Piecewise Projective Homeomorphisms

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Proposition 9. Let A< R be any subring and let p P1 . An a.e. free action of a countable group is amenable in Zim- Then ∈ \{∞} mer’s sense [20, 4.3] if and only if the associated relation is

PSL2(A) p H(A) p. amenable; see [21, Thm. A]. · ⊆ {∞} ∪ · Proof of Theorem 1. Let A = Z be a subring of R. Then Thus, the equivalence relations induced by the actions of 6 ′ 1 A contains a countable subring A < A which is dense in R. PSL (A) and of H(A) on P coincide when restricted to ′ 2 Since H(A ) is a subgroup of H(A), we can assume that A P1 . \ {∞} itself is countable dense. Now H(A) is a countable group and Γ := PSL2(A) is a countable dense subgroup of PSL2(R). Proof. We need to show that given g PSL2(A) with It is proved in Théorème 3 of [22] that the equivalence gp = , there is an element h H(A) such∈ that hp = gp. 6 ∞ ∈ relation on PSL2(R) induced by the multiplication action of We assume g = since otherwise h = g will do. Equiva- Γ is non-amenable; see also Remarks 10 and 11 below. Equiv- lently, we need∞ an 6 ∞ element q G(A) fixing gp and such that −1 ∈ alently, the Γ-action on PSL2(R) is non-amenable. Viewing q = g , writing h = q g. It suffices to find a hyperbolic 1 ∞ ∞ P as a homogeneous space of PSL2(R), it follows that the element q0 PSL2(A) with q0 = g and whose fixed points 1 1 ∈ ∞ ∞ Γ-action on P is non-amenable. Indeed, amenability is pre- ξ± P separate gp from both and g , see Figure 1. In- served under extensions, see [23, 2.4] or [21, Cor. C]. This deed,∈ we can then define q to be the∞ identity∞ on the component 1 action is a.e. free since any non-trivial element has at most of P ξ± containing gp, and define q to coincide with q0 1 \ { } two fixed points. Thus the relation induced by Γ on P on the other component. 1 is non-amenable. Restricting to P , we deduce from Proposition 9 that the relation induced\ {∞} by the H(A)-action is rg also non-amenable. (Amenability is preserved under restric- r ∞ tion [18, 9.3], but here is a null-set anyway.) Thus H(A) ξ+ is a non-amenable group.{∞} r r∞ ξ− Remark 10. We recall from [22] that the non-amenability of the Γ-relation on PSL2(R) is a general consequence of the exis- gpr tence of a non-discrete non-abelian free subgroup of Γ. Thus Fig. 1. The desired configuration of ξ± the main point of our appeal to [22] is the existence of this non-discrete free subgroup, but this is much easier to prove directly in the present case of Γ = PSL2(A) than for general a b non-discrete non-soluble Γ. Let be a matrix representative of g; thus, c d Remark 11. Here is a direct argument avoiding all the above a,b,c,d A and ad bc = 1. The assumption g = references in the examples of A = Z[√2] or A = Z[1/ℓ], where ∈ − ∞ 6 ∞ 1 implies c = 0 and thus we can assume c > 0. Let q0 be ℓ is prime. We show directly that the Γ-action on P is not 6 a b + ra amenable. We consider Γ as a lattice in L := PSL2(R) given by with r A to be determined later; × c d + rc ∈ PSL2(R) in the first case and in L := PSL2(R) PSL2(Qℓ) × thus q0 = g . This matrix is hyperbolic as soon as r is in the second case, both times in such a way that the Γ-action large enough∞ to∞ ensure that the trace τ = a + d + rc is larger| | on P1 extends to the L-action factoring through the first fac- than 2 in absolute value. We only need to show that a suit- tor. If the Γ-action on P1 were amenable, so would be the able choice of r will ensure the above condition on ξ±. Notice L-action (by co-amenability of the lattice). But of course L 1 that and g lie in the same component of P ξ± since does not act amenably since the stabilizer of any point contains ∞ ∞ \ { } q0 preserves these components and sends to g . In con- the (non-amenable) second factor of L. ∞ ∞ clusion, it suffices to prove the following two claims: (1) as The non-discreteness of A was essential in our proof, thus r , the set ξ± converges to ,g ; (2) changing excluding A = Z. |the|→∞ sign of r (when{ r }is large) will change{∞ ∞} the component of 1 | | Problem 12. Is H(Z) amenable? P ,g in which ξ± lie (we need it to be the component of gp\{∞). The∞} claims can be proved by elementary dynamical The group H(Z) is related to Thompson’s group F , for considerations; we shall instead verify them explicitly. which the question of (non-)amenability is a notorious open problem. Indeed F seems to be historically the first candidate The fixed points ξ± are represented by the eigenvectors for a counter-example to the so-called von Neumann conjec- x± , where x± = λ± d rc and where λ± = (τ ture. The relation is as follows: if we modify the definition of c − − ± H(Z) by requiring that the breakpoints be rational, then all 2 1 √τ 4)/2 are the eigenvalues. Now limr→+∞ λ+ = + its elements are automatically C and the resulting group is − ∞ implies limr→+∞ λ− = 0 since λ+λ− = 1 and therefore conjugated to F . The corresponding relation holds between limr→+∞ x− = . Similarly, limr→−∞ x+ =+ (Figure 1 G(Z) and Thompson’s group T . These facts are attributed to depicts the case−∞r > 0). This already proves claim∞ (2) and a remark of Thurston around 1975 and a very detailed expo- half of claim (1). Since g = [a: c], it only remains to verify sition can be found in [24]. ∞ that both limr→+∞ x+ and limr→−∞ x− converge to a, which is a direct computation. H is a free group free group We recall that a measurable equivalence relation with We shall largely follow [16, 3], the main difference being that countable classes is amenable if there is an a.e. defined mea- we replace commutators by§ a non-trivial word in the second surable assignment of a mean on the orbit of each point in such derived subgroup of a free group on two generators. a way that the means of two equivalent points coincide. We refer e.g. to [17] and [18] for background on amenable equiva- The support supp(g) of an element g H denotes the set lence relations. It follows from this definition that any relation p : gp = p , which is a finite union of open∈ intervals. Any produced by a measurable action of a (countable) amenable {subgroup6 of}H fixing some point p P1 has two canonical ∈ group is amenable, by push-forward of the mean [19, 1.6(1)]. homomorphisms to the metabelian stabilizer of p in PSL2(R)

2 www.pnas.org/cgi/doi/10.1073/pnas.0709640104 Footline Author given by left and right germs. Therefore, we deduce the fol- of H(A) which is supported in this neighbourhood. Notice lowing elementary fact, wherein f,g denotes the subgroup that PSL2(Z) contains hyperbolic elements with both fixed h i of H generated by f and g. points ξ± arbitrarily close to , and on the same side. For ∞ Lemma 13. If f,g H have a common fixed point p P1, 2 1 1 n ′′ instance, conjugate by for sufficiently large then any element of∈ the second derived subgroup f,g ∈ acts 1 1 0 1 h i trivially on a neighbourhood of p. n N. We choose such an element h0 with ξ± in the given ∈ Theorem 2 is an immediate consequence of the following neighbourhood and define hS to be trivial on the component 1 more precise statement. of P ξ± containing and to coincide with h0 on the \ { } ∞ Theorem 14. Let f,g H. Either f,g is metabelian or it other component. ∈ h i contains a free abelian group of rank two. A group is called bi-orderable if it carries a bi-invariant Proof. We suppose that f,g is not metabelian, so that total order. The construction below is completely standard, there is a word w in the secondh i derived subgroup of a free compare e.g. [25, p. 233] for a first-order version of our group on two generators such that w(f,g) H is non-trivial. second-order argument. ∈ We now follow faithfully the proof of Theorem 3.2 in [16], Proof of Proposition 6. Choose an orientation of P1 replacing [f,g] by w(f,g). For the reader’s convenience, we and define a (right) germ at a point p to be positive if\{∞} either sketch the argument; the details are on page 495 of [16] (or its first derivative is > 1 or if it is = 1 but the second deriva- [25, p. 232]). Applying Lemma 13 to all endpoints p of the tive is > 0. Then define the set H+ of positive elements of H connected components of supp(f) supp(g), we deduce that ∪ to consist of all transformations whose first non-trivial germ the closure of supp(w(f,g)) is contained in supp(f) supp(g). (starting from along the orientation) is positive. Now H+ is This implies that some element of f,g will send∪ any con- ∞ −1 a conjugacy invariant sub-semigroup and H e is H+ H ; nected component of supp(w(f,g)) toh a disjointi interval. The \{ } ⊔ + this means that H+ defines a bi-invariant total order. needed element might depend on the connected component. Suppose now that we are given a homomorphism from However, upon replacing w(f,g) by another non-trivial ele- ′′ a Kazhdan group to H. Its image is then a Kazhdan sub- ment w1 f,g with minimal number of intersecting com- group K

Footline Author PNAS Issue Date Volume Issue Number 3 n Combining Theorems 1 and 2 with the main result of [36], Hb (H(A),V ) vanishes for all n N and all mixing unitary we conclude that the wreath product Z H is a torsion-free representations V . More generally,∈ it applies to any semi- non-unitarisable group without free subgroups.≀ We can re- separable coefficient module V unless all finitely generated place it by a finitely generated subgroup upon choosing a subgroups of H(A)′′ have invariant vectors in V (see [40] non-amenable finitely generated subgroup of H. This pro- for details and definitions). This should be contrasted with vides some new examples towards Dixmier’s problem, un- the fact that amenability is characterized by the vanishing of solved since 1950 [37, 38, 39]. bounded cohomology with all dual coefficients.

Finally, we mention that our argument from Proposi- ACKNOWLEDGMENTS. Supported in part by the ERC and the Swiss National tion 6.4 in [40] applies to show that the bounded cohomology Science Foundation.

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4 www.pnas.org/cgi/doi/10.1073/pnas.0709640104 Footline Author