Groups of piecewise projective homeomorphisms

Abstract

The group of piecewise projective homeomorphisms of the line provides straightforward torsion-free counterexamples to the so-called von Neumann conjecture. The examples are so simple that many additional properties can be established.

In 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.

Construction

I saw the pale student of unhallowed arts kneeling beside the thing he had put together.

Mary Shelley, introduction to the 1831 edition of Frankenstein

Consider the natural action of the group PSL₂(R) on the projective line P¹ = P¹(R). We endow P¹ with its R-topology, making it a topological circle. We denote by G the group of all homeomorphisms of P¹ that are piecewise in PSL₂(R), each piece being an interval of P¹ with finitely many pieces. We let H < G be the subgroup fixing the point ∞ ∈ P¹ corresponding to the first basis vector of R². Thus, H is left-orderable, because it acts faithfully on the topological line P¹\{∞}, preserving orientations. It follows in particular that H is torsion-free.

Given a subring A < R, we denote by P_A ⊆ P¹ the collection of all fixed points of all hyperbolic elements of PSL₂(A). This set is PSL₂(A)-invariant and countable if A is so. We define G(A) to be the subgroup of G given by all elements that are piecewise in PSL₂(A) with all interval endpoints in P_A. We write H(A) = G(A) ∩ H, which is the stabilizer of ∞ in G(A).

The main result of this article is the following, for which we introduce a method for proving amenability.

Theorem 1.

The group H(A) is nonamenable if A ≠ Z.

The next result is a sequacious generalization of the corresponding theorem of Brin–Squier (16) about piecewise affine transformations, and we claim no originality.

Theorem 2.

The group H does not contain any nonabelian free subgroup. Thus, H(A) inherits this property for any subring A < R.

Thus, already H = H(R) itself is a counterexample to the von Neumann conjecture. Writing H(A) as the directed union of its finitely generated subgroups, we deduce Corollary 3.

Corollary 3.

For A ≠ Z, the groups H(A) contain finitely generated subgroups that are simultaneously nonamenable and without nonabelian free subgroups.

Additional Properties.

The groups H(A) seem to enjoy a number of additional interesting properties, some of which are weaker forms of amenability. In the last section, we shall prove the following five propositions (and recall the terminology). Here, A < R is an arbitrary subring.

Proposition 4.

All L²-Betti numbers of H(A) and G(A) vanish.

Proposition 5.

The group H(A) is inner amenable.

Proposition 6.

The group H is biorderable, and hence, so are all of its subgroups. It follows that there is no nontrivial homomorphism from any Kazhdan group to H.

Proposition 7.

Let E ⊆ P¹ be any subset. Then, the subgroup of H(A), which fixes E pointwise, is coamenable in H(A) unless E is dense (in which case, the subgroup is trivial).

Proposition 8.

If H(A) acts by isometries on any proper CAT(0) space, then either it fixes a point at infinity or it preserves a Euclidean subspace.

One can also check that H(A) satisfies no group law and has vanishing properties in bounded cohomology (see below).

Nonamenability

An obvious difference between the actions of PSL₂(A) and H(A) on P¹ is that the latter group fixes ∞, whereas the former does not. The next proposition shows that this difference is the only one as far as the orbit structure is concerned.

Proposition 9.

Let A < R be any subring, and let p ∈ P¹\{∞}. Then,

Thus, the equivalence relations induced by the actions of PSL₂(A) and H(A) on P¹ coincide when restricted to P¹\{∞}.

Proof.

We need to show that, given g ∈ PSL₂(A) with gp ≠ ∞, there is an element h ∈ H(A), such that hp = gp. We assume g∞ ≠ ∞, because otherwise, h = g will do. Equivalently, we need an element q ∈ G(A) fixing gp and such that q∞ ≠ g∞, writing h ≠ q⁻¹g. It suffices to find a hyperbolic element q₀ ∈ PSL₂(A) with q₀∞ = g∞ and fixed points ξ_± ∈ P¹ that separate gp from both ∞ and g∞ (Fig. 1). Indeed, we can then define q to be the identity on the component of P¹\{ξ_±} containing gp and define q to coincide with q₀ on the other component.

Fig. 1.

The desired configuration of ξ_±.

Let be a matrix representative of g; thus, a, b, c, d, ∈ A, and ad − bc = 1. The assumption g∞ ≠ ∞ implies c ≠ 0, and thus, we can assume c > 0. Let q₀ be given by with r ∈ A to be determined later; thus, q₀∞ ≠ g∞. This matrix is hyperbolic as soon as |r| is large enough to ensure that the trace τ = a + d + rc is larger than 2 in absolute value. We only need to show that a suitable choice of r will ensure the above condition on ξ_±. Notice that ∞ and g∞ lie in the same component of P¹\{ξ_±}, because q₀ preserves these components and sends ∞ to g∞. In conclusion, it suffices to prove the following two claims: (i) as |r| → ∞, the set {ξ_±} converges to {∞, g∞}; and (ii) changing the sign of r (when |r| is large) will change the component of P¹\{∞, g∞} in which ξ_± lie (we need it to be the component of gp). The claims can be proved by elementary dynamical considerations; we shall instead verify them explicitly.

The fixed points ξ_± are represented by the eigenvectors , where x_± = λ_± − d − rc and are the eigenvalues. Now, lim_r→+∞λ₊ = +∞ implies lim_r→+∞λ₋ = 0, because λ₊λ₋ = 1; therefore, lim_r→+∞x₋ = −∞. Similarly, lim_r→−∞x₊ = +∞ (Fig. 1 depicts the case r > 0). Thus, we already proved claim (ii) and half of claim (i). Because g∞ = [a:c], it only remains to verify that both lim_r→+∞x₊ and lim_r→−∞x₋ converge to a, which is a direct computation.□

We recall that a measurable equivalence relation with countable classes is amenable if there is an a.e. defined measurable assignment of a mean on the orbit of each point in such a way that the means of two equivalent points coincide. We refer, e.g., to refs. 17 and 18 for background on amenable equivalence relations. It follows from this definition that any relation produced by a measurable action of a (countable) amenable group is amenable, by push-forward of the mean [ref. 19, 1.6(1)]. An a.e. free action of a countable group is amenable in Zimmer’s sense (ref. 20, 4.3) if and only if the associated relation is amenable (ref. 21, Theorem A).

Proof of Theorem 1.

Let A ≠ Z be a subring of R. Then, A contains a countable subring A′ < A, which is dense in R. Because H(A′) is a subgroup of H(A), we can assume that A itself is countable dense. Now, H(A) is a countable group, and Γ: = PSL₂(A) is a countable dense subgroup of PSL₂(R).

It is proved in Théorème 3 in ref. 22 that the equivalence relation on PSL₂(R) induced by the multiplication action of Γ is nonamenable (see also Remark 10 and Remark 11). Equivalently, the Γ-action on PSL₂(R) is nonamenable. Viewing P¹ as a homogeneous space of PSL₂(R), it follows that the Γ-action on P¹ is nonamenable. Indeed, amenability is preserved under extensions (ref. 21, Corollary C or ref. 23, 2.4). This action is a.e.-free, because any nontrivial element has, at most, two fixed points. Thus, the relation induced by Γ on P¹ is nonamenable. Restricting to P¹\{∞}, we deduce from Proposition 9 that the relation induced by the H(A) action is also nonamenable. [Amenability is preserved under restriction (ref. 18, 9.3), but here, {∞} is a null-set anyway.] Thus, H(A) is a nonamenable group.□

Remark 10.

We recall from ref. 22 that the nonamenability of the Γ-relation on PSL₂(R) is a general consequence of the existence of a nondiscrete nonabelian free subgroup of Γ. Thus, the main point of our appeal to ref. 22 is the existence of this nondiscrete free subgroup, but this existence is much easier to prove directly in the present case of Γ = PSL₂(A) than for general nondiscrete nonsoluble Γ.

Remark 11.

Here is a direct argument avoiding all of the above references in the examples of or A = Z[1/ℓ], where ℓ is prime. We show directly that the Γ-action on P¹ is not amenable. We consider Γ as a lattice in L: = PSL₂(R) × PSL₂(R) in the first case and L: = PSL₂(R) × PSL₂(Q_ℓ) in the second case, both times in such a way that the Γ-action on P¹ extends to the L-action factoring through the first factor. If the Γ-action on P¹ were amenable, so would be the L-action (by coamenability of the lattice). However, of course, L does not act amenably, because the stabilizer of any point contains the (nonamenable) second factor of L.

The nondiscreteness of A was essential in our proof, thus excluding A = Z.

Problem 12.

Is H(Z) amenable?

The group H(Z) is related to Thompson’s group F, for which the question of (non)amenability is a notorious open problem. Indeed, F seems to be historically the first candidate for a counterexample to the so-called von Neumann conjecture. The relation is as follows: if we modify the definition of H(Z) by requiring that the breakpoints be rational, then all its elements are automatically C¹, and the resulting group is conjugated to F. The corresponding relation holds between G(Z) and Thompson’s group T. These facts are attributed to a remark of Thurston around 1975, and a very detailed exposition can be found in ref. 24.

H Is a Free Group Free Group

We shall largely follow ref. 16 (§3), the main difference being that we replace commutators by a nontrivial word in the second derived subgroup of a free group on two generators.

The support supp(g) of an element g ∈ H denotes the set {p: gp ≠ p}, which is a finite union of open intervals. Any subgroup of H fixing some point p ∈ P¹ has two canonical homomorphisms to the metabelian stabilizer of p in PSL₂(R) given by left and right germs. Therefore, we deduce the following elementary fact, wherein 〈f, g〉 denotes the subgroup of H generated by f and g.

Lemma 13.

If f, g ∈ H have a common fixed point p ∈ P¹, then any element of the second derived subgroup 〈f, g〉″ acts trivially on a neighborhood of p.□

Theorem 2 is an immediate consequence of the following more precise statement.

Theorem 14.

Let f, g ∈ H. Either 〈f, g〉 is metabelian or it contains a free abelian group of rank two.

Proof.

We suppose that 〈f, g〉 is not metabelian, so that there is a word w in the second derived subgroup of a free group on two generators such that w(f, g) ∈ H is nontrivial. We now follow faithfully the proof of theorem 3.2 in ref. 16, replacing [f, g] by w(f, g). For the reader’s convenience, we sketch the argument; the details are on p. 495 in ref. 16 or p. 232 in ref. 25. Applying Lemma 13 to all endpoints p of the connected components of supp(f) ∪ supp(g), we deduce that the closure of supp(w(f, g)) is contained in supp(f) ∪ supp(g). This fact implies that some element of 〈f, g〉 will send any connected component of supp(w(f, g)) to a disjoint interval. The needed element might depend on the connected component. However, upon replacing w〈f, g) by another nontrivial element w₁ ∈ 〈f, g〉″ with a minimal number of intersecting components with supp(f) ∪ supp(g), some element h of 〈f, g〉 sends the whole of supp(w₁) to a set disjoint from it. The corresponding conjugate w₂: = hw₁h⁻¹ will commute with w₁, and indeed, these two elements generate freely a free abelian group.□

As pointed out to us by Cornulier, the above argument can be pushed so that w₁ and h generate a wreath product Z ≀ Z (compare ref. 26, Theorem 21 for the piecewise linear case).

Lagniappe

Proof of Proposition 4.

We refer to ref. 27 for the L²-Betti numbers , n ∈ N. Fix a large integer n, and let Γ = G(H) or H(A). Choose a set F ⊆ P_A of n + 1 distinct points, and let Λ < Γ be the pointwise stabilizer of F. Any intersection Λ∗ of any (finite) number of conjugates of Λ is still the pointwise stabilizer of a finite set F∗ containing m ≥ n + 1 points. The definition of G(A) shows that Λ∗ is the product of m infinite groups. The Künneth formula (ref. 27, §2) implies for all i = 0, …, m − 1. In this situation, Theorem 1.3 in ref. 28 asserts for all i ≤ m − 1.□

A subgroup K of a group J is called coamenable if there is a J-invariant mean on J/K. Equivalent characterizations, generalizations, and unexpected examples can be found in refs. 29 and 30.

Recall that a group J is inner amenable if there is a conjugacy-invariant mean on J\{e}. It is equivalent to exhibit such a mean that is invariant under the second derived subgroup J″, because the latter is coamenable in J. Thus, Proposition 5 is a consequence of the stronger fact that H(A) is {asymptotically commutative}-by-metabelian in a sense inspired by ref. 31 as follows.

Proposition 15.

Let A < R be any subring. For any finite set S ⊆ H(A)″, there is a nontrivial element h_S ∈ H(A) commuting with each element of S.

Indeed, any accumulation point of this net of point-masses at h_S is H(A)″-invariant.

Proof of Proposition 15.

By the argument of Lemma 13, there is a neighborhood of ∞ on which all elements of S are trivial. Thus, it suffices to exhibit a nontrivial element h_S of H(A), which is supported in this neighborhood. Notice that PSL₂(Z) contains hyperbolic elements with both fixed points ξ_± arbitrarily close to ∞ and on the same side. For instance, conjugate by for sufficiently large n ∈ N. We choose such an element h₀ with ξ_± in the given neighborhood and define h_S to be trivial on the component of P¹\{ξ_±} containing ∞ and coincide with h₀ on the other component.□

A group is called biorderable if it carries a biinvariant total order. The construction below is completely standard (compare, e.g., ref. 25, p. 233 for a first-order version of our second-order argument).

Proof of Proposition 6.

Choose an orientation of P¹\{∞} and define a (right) germ at a point p to be positive if either its first derivative is >1 or if it is =1 but the second derivative is >0. Then, define the set H₊ of positive elements of H to consist of all transformations with first nontrivial germ (starting from ∞ along the orientation) that is positive. Now, H₊ is a conjugacy invariant subsemigroup, and H\{e} is ; thus, H₊ defines a biinvariant total order.

Suppose now that we are given a homomorphism from a Kazhdan group to H. Its image is then a Kazhdan subgroup K < H. Kazhdan’s property implies that K is finitely generated. It has been known for a long time that any nontrivial finitely generated biorderable group has a nontrivial homomorphism to R: this fact follows ultimately from Hölder’s (32) 1901 work by looking at maximal convex subgroups and is explained in ref. 33 (§2). However, this circumstance is impossible for a Kazhdan group.□

Lemma 16.

For any p ∈ P¹\{∞}, there is a sequence {g_n} in H(Z), such that g_nq converges to ∞ uniformly for q in compact subsets of P¹\{p}.

Proof.

It suffices to show that, for any open neighborhoods U and V of p and ∞, respectively, in P¹, there is g ∈ H(Z), which maps P¹\U into V. Because the collection of pairs of fixed points of hyperbolic elements of PSL₂(Z) is dense in P¹ × P¹, we can find hyperbolic matrices h₁, h₂ ∈ PSL₂(Z) with repelling fixed points r_i in U\{p} and attracting fixed points a_i in V\{∞}, such that the cyclic order is ∞, a₁, r₁, p, r₂, a₂. Now, we define g to be a sufficiently high power of h₁ on the interval [a₁, r₁] (for the above cyclic order), h₂ on the interval [r₂, a₂], and the identity elsewhere.□

Proof of Proposition 7.

Let K be the pointwise stabilizer of a nondense subset E ⊆ P¹; it suffices to find a mean invariant under H(A)″. Let {g_n} be the sequence provided by Lemma 16 for p, an interior point of the complement of E. Any accumulation point of the sequence of point-masses at g_nK in H(A)/K will do. Indeed, because any g ∈ H(A)″ is trivial in a neighborhood of ∞, we have for n large enough.□

The existence of two (or more) commuting coamenable subgroups is also a weak form of amenability. It is the key in the argument cited below.

Proof of Proposition 8.

Consider two disjoint nonempty open sets in P¹. The pointwise stabilizers of their complement commute with each other and are coamenable by Proposition 7. In this situation, Corollary 2.2 in ref. 34 yields the desired conclusion.□

The properties used in this section show immediately that H(A) fulfills the criterion of ref. 35, Theorem 1.1 and thus, satisfies no group law.

Combining Theorems 1 and 2 with the main result of ref. 36, we conclude that the wreath product Z ≀ H is a torsion-free nonunitarisable group without free subgroups. We can replace it by a finitely generated subgroup upon choosing a nonamenable, finitely generated subgroup of H. This construction provides some new examples to Dixmier’s problem, unsolved since 1950 (37–39).

Finally, we mention that our argument from Proposition 6.4 in ref. 40 applies to show that the bounded cohomology vanishes for all n ∈ N and all mixing unitary representations V. More generally, it applies to any semiseparable coefficient module V, unless all finitely generated subgroups of H(A)″ have invariant vectors in V (see ref. 40 for details and definitions). This vanishing should be contrasted with the fact that amenability is characterized by the vanishing of bounded cohomology with all dual coefficients.