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Bernoulli actions of amenable groups with weakly mixing Maharam extensions

We provide a simple criterion for a non-singular and conservative Bernouilli action to have a weakly mixing Maharam extension. As an application, we show that every countable amenable group admits a stable type III_1 Bernoulli action, answering a recent question by Vaes and Wahl.


The Maharam extension
Let G be a countable group and let (X, µ) be a Borel G-space, that is to say, a Borel probability measure space (X, µ) endowed with an action of G by measure-class preserving measurable maps. It is well-known that there exists a µ-conull G-invariant subset X ′ ⊂ X such that the Radon-Nikodym cocycle is defined for all (g, x) ∈ G × X ′ . The Maharam extension G ( X, µ) associated to (X, µ) is the G-action on the set X = X ′ × R defined by g(x, t) = (gx, t + r µ (g, x)), for (x, t) ∈ X ′ × R, which can be readily checked to preserve the infinite measure µ = µ ⊗ e t dt.
We say that the Maharam extension of (X, µ) is weakly mixing if for every probability measurepreserving ergodic G-space (Y, ν), the diagonal action G ( X × Y, µ ⊗ ν) is ergodic, or equivalently, if the Maharam extension of G (X × Y, µ ⊗ ν) is ergodic. It is not hard to see that if the Maharam extension of (X, µ) is ergodic, then there is no G-invariant σ-finite measure equivalent to µ. Borel G-spaces with weakly mixing Maharam extensions are often referred to as being of stable type III 1 in the literature.
In this paper we shall primarily be interested in the case when the action is conservative, ergodic and there is no G-invariant σ-finite measure equivalent to µ. Such systems are often 1 said to be of of type III, and the first examples of type III-Bernoulli actions for G = (Z, +) were constructed by Hamachi in [8]. As it turns out, ergodic type III-actions of any group decompose further into a one-parameter family of orbit equivalence classes III λ , where 0 λ 1, and being of type III 1 is equivalent to having an ergodic Maharam extension (see for instance [10] for more details). Ulrich Krengel and Benjamin Weiss asked early on: Which III λ -types can occur among non-singular Bernoulli actions? Despite the classical flair of this question, the first examples of non-singular Bernoulli actions of G = (Z, +) with ergodic Maharam extensions, and thus type III 1 , were constructed by the second author [11] as late as 2009. A few years later, the second author [12] further showed that for a large class of Bernoulli actions of G = (Z, +), being of type III implies that the Maharam extensions are weakly mixing (in fact, have the K-property). In particular, this shows that within this class of examples, only type III 1 is possible. Danilenko and Lemańczyk extended this in [5] to a more general class of Bernoulli actions. However, all of these examples are rather special to the additive group of integers as they rely on the notion of a "past" in the group.
More recently, Vaes and Wahl in [14] studied the question above for general countable groups. In their paper, they formulate and make substantial progress on, the following conjecture: Conjecture. [14] Let G be a countable group with non-vanishing first ℓ 2 -cohomology, or equivalently, suppose that there is a function f : Then there is a non-singular Bernoulli action G ({0, 1} G , µ) with an ergodic (or even weakly mixing) Maharam extension.
If G does not admit a function as above, for instance, if G has property (T), then they show [14, Theorem 3.1] that every non-singular and conservative Bernoulli action of G is equivalent to a probability-measure preserving one, so in particular it cannot have an ergodic Maharam extension. On the other hand, amenable groups do admit plenty of such functions, and in this case they show: They further show that if one is willing to replace the set {0, 1} with a Cantor set Z, then every countable amenable group admits a non-singular Bernoulli action G (Z G , µ) with a weakly mixing Maharam extension. They explicitly ask whether the passage to a Cantor set is necessary, or whether one can keep the base {0, 1} in this generality. One of the aims of this paper is to prove that one always can.

Main results
Crucial to the approach of Vaes and Wahl is a two-part criterion [14, Proposition 6.6] for when a non-singular Bernoulli action of a countable amenable group G has a weakly mixing Maharam extension, which involves a simple asymptotic condition on the family (µ g ) on {0, 1} and a rather technical assumption that G can be exhausted by so called "λ-inessential" subsets. The conditions on G in Theorem 1.1 are there to guarantee that such an exhaustion is possible.
We show in this paper that the latter technical condition is not needed, and that the asymptotic condition on (µ g ) stated by Vaes and Wahl, together with the conservativity of the resulting Bernoulli action, is enough to guarantee that the Maharam extension is weakly mixing. More precisely, we shall show: Theorem 1.2. Let G be a countable amenable group and suppose that G ({0, 1} G , µ) is a non-singular conservative Bernoulli action with the property that there exist δ > 0 such that δ µ g (0) 1 − δ for all g ∈ G and a probability measure λ on {0, 1} such that lim g→∞ µ g (0) = λ(0) and Then the Maharam extension of G ({0, 1} G , µ) is weakly mixing.

Remark 1.3.
Conversely, we prove in the appendix that if a countable group G admits a nonsingular Bernoulli action which satisfies the conditions (1.2), then G must be amenable.
Vaes and Wahl [14, Proposition 4.1 and 6.8] prove that every countable amenable group admits a non-singular and conservative Bernoulli action satisfying the conditions (1.2) with λ(0) = λ(1) = 1/2 (this is not quite the way it is phrased in [14]; see remark below). In combination with Theorem 1.2, this allows us to conclude that:

A few words about the proof of Theorem 1.2
Let G be a countable amenable group and G (X, µ) = ({0, 1} G , µ) a non-singular and conservative Bernoulli action satisfying the conditions of Theorem 1.2. In particular, if we set then we see that either (1)), for all g, we may without loss of generality assume that the first alternative holds (otherwise we just interchange 0 and 1). So, from now on, the standing assumptions throughout the rest of the paper are: Let now (Y, ν) be an ergodic probability-measure-preserving G-space. By the recent result of Danilenko [6, Theorem 0.1], the diagonal action G (X × Y, µ ⊗ ν) is ergodic, and we wish to prove that its Maharam extension is ergodic as well. By [13,Corollary 5.4], this is equivalent to showing that for every ε > 0 and t 0 (or t 0) and µ ⊗ ν-measurable subset C ⊂ X × Y with positive measure, Remark 1.6. A word of clarification might be in order. In Corollary 5.4 in [13], to prove ergodicity, one needs to ensure (1.5) for all t ∈ R. However, by [13,Lemma 3.3], the set of t for which (1.5) holds is a closed subgroup of (R, +), so in particular symmetric.
Let us fix a dense subset A of the measure algebra of (X × Y, µ ⊗ ν). By [3, Lemma 2.2], to prove (1.5) for every µ⊗ ν-measurable subset of X × Y with positive measure, it suffices to show that there exists M 1 with the property that for ε > 0 and t 0 (or t 0) and C ∈ A, we have Let A denote the subset of the measure algebra of (X × Y, µ ⊗ ν) consisting of finite unions of product sets of the form Borel sets. Then A is dense, and to prove (1.6) for every set in A, it suffices by disjointness of the sets A i × B i , to show that there is a constant M such that (1.6) holds for every set of the form C = A i × B i . This is the content of the following proposition which is our main technical result, and whose proof will occupy the rest of the paper. Proposition 1.7. Fix ε > 0, a cylinder set A ⊂ X and a Borel set B ⊂ Y.

Acknowledgements
The first author would like to thank the Einstein Institute of Mathematics at Hebrew University in Jerusalem for the hospitality during the time this paper was written.

PROOF OF PROPOSITION 1.7
Let G be a countable amenable group and let (X, µ) = ({0, 1} G , µ) and λ be as in Theorem 1.2. We also fix a probability measure-preserving ergodic G-space (Y, ν).

Shapes of cylinder sets
If K ⊂ G is a finite set and σ : K → {0, 1}, we define the cylinder set A with shape(A) = (K, σ) by

The homoclinic relation and its cocycles
We define the homoclinic relation H µ on X ′ by g , for all but finitely many g . For any family (η g ) of functions on {0, 1} we can define the homoclinic cocycle (or Gibbs cocycle) associated to this family by The case when η g (a) = log µ g (a) λ(a) for a ∈ {0, 1} will be of special importance to us, and to distinguish this case, we omit η as an index. In other words, we write We denote by [H µ ] the collection of partially defined one-to-one measurable maps between measurable subsets of X ′ whose graphs are subsets of H µ . If φ ∈ [H µ ], we write Dom(φ) for its domain of definition, and set We note that if A ⊂ X is a cylinder set with shape(A) = (K, σ) and K ∩ supp(φ) = ∅, then and µ(A ∩ Dom(φ)) = µ(A)µ(Dom(φ)).

Three lemmas
In what follows, we shall show how Proposition 1.7 can be redued to three lemmas, whose proofs are postponed to the next sections.
Our first lemma roughly says that the cocycle c defined in (2.2) attains "most values" quite frequently, and its proof is very much inspired by [ and Our second lemma provides a way to compare the cocycles r µ and c, and roughly says that if φ ∈ [H µ ] and r µ (g, φ(x)) is small and c(x, φ(x)) is close to t, then r µ (g, x) is close to t as well.

Lemma 2.2. For every
Our third lemma is very general and says that if G (X × Y, µ ⊗ ν) is conservative, then the cocycle r µ⊗ν (g, ·) is "frequently" very small for arbitrarily large g.

Lemma 2.3.
For every ε > 0 and finite subset F ⊂ G and µ ⊗ ν-measurable set C ⊂ X × Y, we have
If η g : {0, 1} → R is a sequence of functions, indexed by g ∈ G, we define the associated homoclinic (or Gibbs) cocycle c η : H µ → R by

Proof of Lemma 3.1
For g ∈ G, we define τ g : X → X by We first write c η on the form where F g : X → R are defined as We note that all (F g ) are independent of each other, and Let us fix ε > 0 and choose a finite subset M ⊃ K such that F g ∞ < ε/2 for all g / ∈ M. We enumerate G + \ M = {g 1 , g 2 , . . .} and set G n = {g 1 , . . . , g n } and S n = g∈G n F g and A n = g∈G n X F g dµ.
Since δ µ g (0) 1 − δ for all g ∈ G, it is not hard to see that there is a constant C δ such that for all n, and thus the variance B 2 n := X |S n − A n | 2 dµ tends to infinity as n → ∞ by (3.3). Hence, by Lyapunov's CLT (see e.g. [4,Theorem 7 for all r ∈ R. In particular, there exist n + , n − 1 such that µ x ∈ X : S n (x) > A n = µ x ∈ X : S n (x) − A n B n > 0 1 3 , for all n n + and µ x ∈ X : S n (x) < A n = µ x ∈ X : S n (x) − A n B n < 0 1 3 , for all n n − .
If (3.4) holds, then A n → ∞, so for any t 0, the inequality A n t holds eventually, and thus we can find N + = N + (t) n + such that Let us from now on fix t 0 and an integer N + = N + (t) as above, and set and φ + :
If (3.5) holds, then A n → −∞, so for any t 0, the inequality A n t holds eventually, and thus we can find N − = N − (t) n − such that Let us from now on fix t 0 and an integer N − = N − (t) as above, and set Clearly, The rest of the argument is now completely analogous to previous case.

Proof of Lemma 2.2
We recall the definitions of the cocycles r µ and c from Section 2. In particular, by (2.1), we know that for all x ∈ X ′ , and for all (x, x ′ ) ∈ H µ , . Fix ε > 0 and φ ∈ [H ν ] with finite support and set Since δ µ g (0) 1 − δ for all g ∈ G and µ g (0) → λ(0) as g → ∞, the set L is finite. We note that for every g ∈ G and x ∈ Dom(φ), If L ∩ g(supp(φ)) = ∅, then the absolute value of the last sum is bounded by ε, which finishes the proof.

PROOF OF LEMMA 2.3
We retain the notation from Section 2. It is not hard to see that G (X, µ) essentially free. Since we have assumed that it is also conservative, so is the diagonal action G (X × Y, µ ⊗ ν). Hence Lemma 2.3 follows from the following general result, which is surely known to experts, but we include it for completeness.  Proof. Since G (Z, ξ) is conservative, it follows from [13, Theorem 4.2 and Theorem 5.5] that its Maharam extension G ( Z, ξ) is conservative as well. Since G (Z, ξ) is essentially free, so is G ( Z, ξ). Hence by Halmos recurrence theorem [1, Proposition 1.6.2], for every measurable set C ⊂ Z, we have D = C modulo ξ-null sets, where Let δ > 0 and fix a Borel set C ⊂ Z. If we define C = C × (−δ, δ) and let D denote the corresponding set defined in (4.2), then Fubini's Theorem tells us that there is a ξ-conull subset C ′ ⊂ C such that D z := t ∈ R : (z, t) ∈ D = (−δ, δ), modulo ξ-null sets, for all z ∈ C ′ . In particular, for all z ∈ C ′ , we have D z g∈G χ C (gz, t + r ξ (g, z)) e t dt = g∈G χ C (gz)ρ δ (r ξ (g, z)) = ∞, where ρ δ (s) = δ −δ χ (−δ,δ) (s + t)e t dt, for s ∈ R.

APPENDIX A. ON THE ROLE OF AMENABILITY
Proposition A.1. Let G be a finitely generated non-amenable group and suppose that G ({0, 1} G , µ) is a non-singular Bernoulli action with the property that there exist δ > 0 such that δ µ g (0) 1 − δ for all g ∈ G and a probability measure λ on {0, 1} such that lim g→∞ µ g (0) = λ(0).
Then µ is equivalent to the G-invariant probability measure g λ. In particular, it is not of type III.
Remark A.2. The proof below can be adapted to also deal with infinitely generated groups, but providing references in this generality would become harder.
Let S be a finite and symmetric generating set of G. Since G is non-amenable, we can, by [2, Corollary 3], write µ g (0) = u(g) + v(g), for all g ∈ G, where v ∈ ℓ 2 (G) and 1 |S| s∈S u(gh) = u(g), for all g ∈ G. (A.1) Since v ∈ ℓ 2 (G) and G is infinite, we must have lim g v(g) = 0, whence lim g u(g) = λ(0). There are many ways to see why u has to be a constant (and thus equal to λ(0)). For instance, let (z n ) be the simple random walk on G with steps in S. Then, z n → ∞ almost surely (since G is non-amenable), and thus u(gz n ) → λ(0) for all g ∈ G. On the other hand, by (A.1) and the dominated convergence theorem, u(g) = lim n E[u(gz n )] = λ(0), for all g ∈ G.