Relative ergodicity results for product groups

May 7, 2006

Contents

1 Introduction 1 2 Definitions 1 3 Relative ergodicity result 3 4 Amenable classes of structures 11 5 Non-reduction 15 A The boundary of the free group 23 B Skew invariant measures 26 C E_{0}-ergodicity 29

1 Introduction

These notes present various relative ergodicity results, to the effect that actions of Bernoulli shifts of a comparatively larger number of copies of the free group must be relatively ergodic to free actions of smaller numbers of copies of the free group. Ultimately this is used to give a “quote-unquote” elementary proof of a theorem previously established by Adams and Kechris, that there are ≤_{B}-incomparable countable Borel equivalence relations.

2 Definitions

Definition We let E_{0} be the equivalence relation of eventual agreement on infinite binary sequences; thus for f, g ∈ {0, 1}^{ℕ} we set

fE_{0}g

if there is some N such that for all n > N

f(n) = g(n).

Definition Given groups standard Borel groups G, H, with G acting in a Borel manner on standard Borel X, we say that

α : X × G → H

is a cocycle if it is Borel and for all x ∈ X, g_{1}, g_{2} ∈ G

α(x, g_{1}g_{2}) = α(g_{2} · x, g_{1})α(g_{1}, x).

Two cocycles

α, β : X × G → H

are cohomologous if there is a Borel

ψ : X → H

such that for all x ∈ X, g ∈ G

β(x, g) = ψ(g · x)α(x, g)ψ(x)^{−1}.

Frequently we will have a Borel probability measure μ on X and we can talk about a μ-measurable cocycle, which will be a cocycle defined on some conull subset of (X, μ), and two cocycles being μ-measurably cohomologous if they are outright cohomologous on a conull set. When there is no ambiguity about the measure, we will simply write measurable in place of the more lengthy μ-measurable.

In the present paper we will only really be considering the case of G and H both countable. Admittedly many of the results extend to the wider context of locally compact.

Definition Let E and F be equivalence relations on standard Borel X and Y. A Borel function

ρ : X → Y

is a homomorphism of E to F if for all x_{1}, x_{2} ∈ X

x_{1}Ex_{2} ⇒ ρ(x_{1})F ρ(x_{2});

in other words, ρ × ρ[E] ⊂ F. An equivalence relation E on a standard Borel probability space (X, μ) is said to be F -ergodic if whenever ρ is a homomorphism from E to F then there is a single y_{0} such that for μ-a.e. x ∈ X we have ρ(x)F y_{0}; in other words, [ρ(x)]_{F} is constant on a conull set.

We write E ≤_{B} F, E is Borel reducible to F, if there is a homomorphism ρ from E to F which has the stronger property that x_{1}Ex_{2} if and only if ρ(x_{1})F ρ(x_{2}); in other words, ρ can be thought of as pushing down to an injection between the quotient spaces,

X/E ↪ Y/F.

It is known that an equivalence relation with all classes countable is hyperfinite, that is to say writeable as an increasing union of Borel equivalence relations with finite classes, if and only if it is Borel reducible to E_{0}.

At some point it will be convenient to consider classes of reductions somewhat broader than Borel. For instance, we say that E is universally measurably reducible to F if there is a universally measurable

ρ : X → Y

such that for all x_{1}, x_{2} ∈ X

x_{1}Ex_{2} ⇔ ρ(x_{1})F ρ(x_{2}).

Among the countable (that is to say, all classes are countable) Borel equivalence relations there are maximal ones with respect to the ordering of Borel reducibility. We arbitrarily choose one of these, and denote it by E_{∞} and assume it lives on the standard Borel space X_{∞}. While it is immediate from the definitions that any countable Borel equivalence relations E must be Borel reducible to E_{∞}, it is not true that E ≤_{B} E_{∞} implies E is countable; in this case we say that E is essentially countable.

Proofs of these and related facts can be found in [6] and [7].

A homomorphism ρ from E on X to F on Y is said to factor through a third equivalence relation F′ on Y′ if there are homomorphisms

ρ_{1} : X → Y′,
ρ_{2} : Y′ → Y,

from E to F′ and from F′ to F, with

ρ = ρ_{2} ∘ ρ_{1}.

An important example of a cocycle arises from a homomorphism of one orbit equivalence relation into another. Suppose G acts in a Borel manner on X and H acts freely and in a Borel manner on Y and

ρ : X → Y

is a homomorphism of E_{G} to E_{H}. Then we obtain an induced cocyle

α : X × G → H

by letting

α(x, g) = h

if h ∈ H is the unique group element with

h · ρ(x) = ρ(g · x).

The characterstic identity of the definition of a cocyle is easily checked. We do need to argue that α(·, ·) is indeed Borel, but this follows by the classical uniformization theorem for Borel sets in the plane with countable sections (see for instance the relevant passages of [10]).

3 Relative ergodicity result

Lemma 3.1 Let G and H be countable groups. Suppose G acts by measure preserving transformations on standard Borel (X, μ) and H acts freely by Borel automorphisms on standard Borel Y. Suppose that the induced cocycle

α : X × G → H

is cohomologous to a cocycle into an abelian subgroup of H and suppose further that E_{G} is E_{0}-ergodic. Then there is a single y_{0} ∈ Y such that for μ-a.e. x ∈ X we have

ρ(x)E_{H}y_{0}.

Proof Restating the assumptions of the lemma gives us some abelian H_{0} < H and Borel

ψ : X → H

such that for all x ∈ X, g ∈ G

ψ(g · x)α(x, g)ψ(x)^{−1} ∈ H_{0}.

Therefore if we define

ρ_{0} : X → Y
x ↦ ψ(x) · ρ(x),

then for any x, g and

h_{0} = ψ(g · x)α(x, g)ψ(x)^{−1}

we must have

h_{0} · ρ_{0}(x) = ψ(g · x)α(x, g)ψ(x)^{−1}ρ_{0}(x)
= ψ(g · x)α(x, g)ψ(x)^{−1}ψ(x)ρ(x) = ψ(g · x)α(x, g)ρ(x)
= ψ(g · x)ρ(g · x)

by definition of α(·, ·), which in turn equals

ρ_{0}(g · x).

Thus the induced cocycle for the new homomorphism ρ_{0} of E_{G} to E_{H} always takes its values inside H_{0}.

In other words, ρ_{0} is in fact a homomorphism from E_{G} to E_{H_{0}}, from the orbit equivalence relation induced by the action of G on X to the equivalence relation induced by the action of H_{0} on Y.

We can accordingly define a measure

ν = (ρ_{0})_{∗}(μ)

on Y by

ν(B) = μ(ρ_{0}^{−1}[B]).

As remarked in [7], it follows from [2] that any countable abelian group of Borel automorphisms gives rise to an equivalence relation which is hyperfinite, and thus Borel reducible to E_{0}. In particular, E_{H_{0}} ≤_{B} E_{0}.

The lemma now follows by E_{0}-ergodicity of E_{G}. □

Lemma 3.2 Suppose G and H are countable groups and K ◁ G is normal. Suppose that G acts by measure preserving transformations on probability space (X, μ) and the action of K is by itself already ergodic.

Suppose that

α : X × G → H

is a cocycle with

α(x, k) = 1

for all k ∈ K and μ-a.e. x ∈ X.

Then there is a homomorphism

π : G → H

such that for all g ∈ G and μ-a.e. x ∈ X

α(x, g) = π(g).

Proof For a.e. x and any g ∈ G, k ∈ K we have

α(k · x, g) = α(k · x, g)α(x, k)
= α(x, gk) = α(x, k̄g),

for k̄ = gkg^{−1} ∈ K, which in turn on a conull set equals

α(g · x, k̄)α(x, g) = α(x, g).

Thus the function

x ↦ α(x, g)

is K-invariant, and so by the ergodicity of K constant almost everywhere.

If we define π(g) to be the almost everywhere constant value of α(x, g) then we obtain for any g_{1}, g_{2} ∈ G that for a conull set of x ∈ X

π(g_{1}g_{2}) = α(x, g_{1}g_{2})
= α(g_{2} · x, g_{1})α(x, g_{2}) = π(g_{1})π(g_{2}).
□

As a word towards notation, if G is a discrete group and S a discrete set, we let {0, 1}^{G×S} be the collection of functions

f : G × S → {0, 1}

equipped with the product topology. We allow G to act on this space of functions by

(σ · f)(τ, s) = f(σ^{−1}τ, s).

Theorem 3.3 Let S be a countable set. Let G = (𝔽_{2})^{n} × ℤ act freely on

X =_{df} {0, 1}^{G×S} = ∏_{G×S}{0, 1}

by shift; view X as coming equipped with the usual G-invariant product measure.

Let m < n and suppose that H = (𝔽_{2})^{m} × ℤ acts freely by Borel automorphisms on standard Borel Y. Then E_{G} is E_{H}-ergodic.

Proof We prove this by induction on n, with the base case n = 1 following by 3.1 and appendix C. Assuming now that n > 1 let us suppose the theorem is true for all n′ < n.

At each i ≤ n, j ≤ m let

G_{i} = {1} × {1} × ...𝔽_{2}^{(i^{th}place)} × {1}... × {0} ◁ G,

the i^{th} copy of the free group in G,

H_{j} = {1} × {1} × ...𝔽_{2}^{(j^{th}place)} × {1}... × {0} ◁ H,

the j^{th} copy of the free group in H, and let

K = {1} × {1}... × {1} × ℤ ◁ G.

We take

β : X × G → H

to be the induced cocyle, and at each i ≤ n we let

β_{i} : X × G_{i} → H

be the restriction β|_{X×G_{i}}. Defining at each j ≤ m

π_{j} : H ↠ H_{j}

to be the canonical surjection, we let

β_{i,j} : X × G_{i} → H_{j}

to be the cocycle given by

β_{i,j} = π_{j} ∘ α_{i}.

We similarly let

β̂ : X × K → H

be the cocycle induced for K and let β̂_{j} : X × K → H_{j} be defined by β̂_{j} = π_{j} ∘ β̂.

For now on let us fix some i ≤ n. Case 1 At each j ≤ m we have that β_{i,j} is measurably cohomologous to a cocycle into an abelian subgroup of H_{j}.

Then at each j fix measurable

ψ_{j} : X → H_{j}

and abelian H_{j}^{0} < H_{j} such that

ψ_{j}(g · x)α_{i,j}(x, g)ψ_{j}(x)^{−1} ∈ H_{j}^{0}

for all g ∈ G_{i} and almost every x ∈ X. Thus if we define

ψ⃗ : X → H
x ↦ ψ_{1}(x)ρ_{2}(x) ⋯ ψ_{m}(x)

then we have that

ψ⃗(g · x)β_{i}(x, g)ψ⃗(x)^{−1} ∈ H_{1}^{0} × H_{2}^{0} × ⋯ H_{m}^{0} × Z.

In particular, β_{i}(·, ·) is (almost everywhere) cohomologous to a cocycle into an abelian subgroup of H. With this established, the lemma follows by 3.1 and appendix C.

Case 2 There is some j ≤ m with β_{i,j} not measurably cohomologous to a cocycle into an abelian subgroup of H_{j}.

From now until the finish we fix this j. The next argument parallels a similar point in [11]. Claim: We can find a measurable assignment

x ↦ μ_{x}
X → M_{1,3}^{+}(H_{j})

with

β̂_{j}(x, k) · μ_{x} = μ_{k·x}

for all k ∈ K and a.e. x ∈ X. Proof of claim: Following appendix B we can certainly find a measurable assignment

x ↦ μ_{x}
X → M_{1}^{+}(∂H_{j})

with

β̂_{j}(x, k) · μ_{x} = μ_{k·x}

for all k ∈ K and a.e. x ∈ X. Let us assume for a contradiction that we cannot find such an assignment with the measures not just concentrating on at most 2 ends almost everywhere. Appealing to the ergodicity of the action of K on X we have that either almost everywhere μ_{x} concentrates on two ends or almost everywhere the measure concentrates on exactly one measure. Which ever case holds, we may assume that there is no other assignment concentrating on a greater number of ends.

For organizational simplicity, we will simply assume that μ_{x} almost everywhere concentrates on two ends, \{e_{x}, e′_{x}\}; without any loss we can further simplify the situation by assuming μ_{x}(\{e_{x}\}) = μ_{x}(\{e′_{x}\}) = \frac{1}{2}. We then have that

{β̂_{j}(x, k) · e_{x}, β̂_{j}(x, k) · e′_{x}} = {e_{k·x}, e′_{k·x}}

all k, a.e. x, and that for any other such assignment

X → [∂H_{j}]^{2}
x ↦ {f_{x}, f′_{x}}

with {β̂_{j}(x, k) · f_{x}, β̂_{j}(x, k) · f′_{x}} = {f_{k·x}, f′_{k·x}} all k, a.e. x we have

{e_{x}, e′_{x}} = {f_{x}, f′_{x}}

almost everywhere. Now note the following:

For any g ∈ G_{i}, if we let

{f_{x}, f′_{x}} = {β_{i,j}(x, g)^{−1}e_{g·x}, β_{i,j}(x, g)^{−1}e′_{g·x}}

then for all k ∈ K and a.e. x ∈ X

{β̂_{j}(x, k) · f_{x}, β̂_{j}(x, k) · f′_{x}} = {f_{k·x}, f′_{k·x}}.

To see this we use that gk = kg and appeal to the defining identity of cocycles to obtain

β(x, gk) = β(x, kg)
∴ β(k · x, g)β(x, k) = β(g · x, k)β(x, g)
∴ β(k · x, g)β(x, k)β(x, g)^{−1} = β(g · x, k)
∴ β(x, k)β(x, g)^{−1} = β(k · x, g)^{−1}β(g · x, k),

which in particular gives

β̂_{j}(x, k)β_{i,j}(x, g)^{−1} = β_{i,j}(k · x, g)^{−1}β̂_{j}(g · x, k),

and hence

{β̂_{j}(x, k) · f_{x}, β̂_{j}(x, k) · f′_{x}} =_{df} {β̂_{j}(x, k)β_{i,j}(x, g)^{−1}e_{g·x}, β̂_{j}(x, k)β_{i,j}(x, g)^{−1}e′_{g·x}}
= {β(k · x, g)^{−1}β(g · x, k) · e_{g·x}, β(k · x, g)^{−1}β(g · x, k) · e′_{g·x}}
= {β(k · x, g)^{−1} · e_{kg·x}, β(k · x, g)^{−1} · e′_{kg·x}}
= {β(k · x, g)^{−1} · e_{g·(k·x)}, β(k · x, g)^{−1} · e′_{g·(k·x)}} =_{df} {f_{k·x}, f′_{k·x}}.

That point then granted, we have by our assumptions on {e_{x}, e′_{x}} that

{e_{x}, e′_{x}} = {f_{x}, f′_{x}}

almost everywhere. Since our choice of g ∈ G_{i} was arbitrary, we have discovered that for every g ∈ G_{i} that

μ_{x} = β_{i,j}(x, g)^{−1}μ_{g·x};

in other words,

μ_{g·x} = β_{i,j}(x, g)μ_{x}.

In particular the map

x ↦ μ_{x}
X → M_{1}^{+}(∂𝔽_{2})

is a homomorphism from E_{G_{i}}^{X} to the equivalence relation E_{H_{j}}^{M_{1}^{+}(∂H_{j})}. From A.6 the latter equivalence relation is hyperfinite, and then from C we have therefore that there is a single μ̂ ∈ M_{1}^{+}(∂H_{j}) such that for almost all x ∈ X

μ_{x}F_{H_{j}}^{M_{1}^{+}(∂H_{j})}μ̂.

Thus we may in a Borel manner assign a group element h_{x} to each x such that almost everywhere

h_{x} · μ_{x} = μ̂.

Thus if we replace β_{i,j}(·, ·) by the cohomologous

β̄ : X × G_{i} → H_{j}
(x, g) ↦ h_{g·x}β_{i,j}(x, g)h_{x}^{−1}

then for all g and a.e. x

β̄(x, g) · μ̂ = h_{g·x}β_{i,j}(x, g)h_{x}^{−1} · μ̂
= h_{g·x}β_{i,j}(x, g) · μ_{x} = h_{g·x} · μ_{g·x} = μ̂.

Thus almost everywhere β̄ maps into the stabilizer of μ̂, which from A.6 must be abelian. With a contradiction to the assumption on i and j. (□Claim)

We can now quote A.1 and find a Borel

s : M_{1,3}^{+}(∂H_{j}) → H_{j}

such that s(h · μ) = hs(μ), h ∈ H_{j}, μ ∈ M_{1,3}^{+}(∂H_{j}), and then appealing to the claim we define measurable

ŝ: X → H_{j}
x ↦ s(μ_{x}).

We have for all k ∈ K and a.e. x ∈ X

ŝ(k · x) = s(μ_{k·x}) = s(β̂_{j}(x, 2) · μ_{x})
= β̂_{j}(x, k) · s(μ_{x}) = β̂_{j}(x, k) · ŝ(x)
∴ ŝ(k · x)^{−1}β̂_{j}(x, k)ŝ(x) = 1.

Thus let us replace the homomorphism

ρ : X → Y

by

ρ̂: X → Y
x ↦ ŝ^{−1} · ρ(x);

then we have a new homomorphism from E_{G}^{X} to E_{H}^{Y} which has the property that ρ̂(x) = ρ(x) almost everywhere. Consequently it suffices to show to prove the theorem for this new homomorphism ρ̂. By analogy with before we let

γ : X × G → H

to be the induced cocyle, and let

γ_{j} : X × G → H_{j}

be defined by γ_{j} = π_{j} ∘ γ. Claim: γ_{j}|_{X×K} is trivial almost everywhere.

Note then for almost all x, all k ∈ K, if we choose h⃗ = γ(x, k) then

h⃗ · ρ̂(x) = ρ̂(k · x),

∴ h⃗ · ŝ(x)^{−1} · ρ(x) = ŝ(k · x)^{−1}ρ(k · x)
∴ ŝ(k · x)h⃗ · ŝ(x)^{−1} · ρ(x) = ρ(k · x)

which implies that

ŝ(k · x)h⃗ · ŝ(x)^{−1} = β̂(x, k)
∴ π_{j}(h⃗) = π_{j}(ŝ(k · x)^{−1}β̂(x, k)ŝ(x))
= ŝ(k · x)^{−1}β̂_{j}(x, k)ŝ(x) = 1.
(□Claim)

Thus by 3.2 we obtain an actual homomorphism

π̂ : G → H_{j}

such that almost every where

γ_{j}(x, g) = π̂(g).

We want to show that π̂[G_{i′}] = {1} for i′ ≠ i. For this purpose it suffices to prove the following claim: Claim: If σ_{1}, σ_{2}, σ_{3} ∈ 𝔽_{2}, all non-trivial, and σ_{1} and σ_{2} commute as do σ_{1} and σ_{3}, then σ_{2} and σ_{3} commute as well. Proof of claim: Since subgroups of free groups are free, we have ⟨σ_{1}, σ_{2}⟩ ≅ ℤ, and thus we can find some wτw^{−1} such that

⟨wτw^{−1}⟩ ⊃ ⟨σ_{1}, σ_{2}⟩.

We may assume by taking τ as short as possible that τ ≠ σ^{k} for any k > 1, σ ∈ 𝔽_{2}; and then we may assume that τ ≠ vσv^{−1} any σ ∈ 𝔽_{2} and v ≠ 1. Thus there will be ℓ_{1}, ℓ_{2} with

σ_{1} = wτ^{ℓ_{1}}w^{−1},

σ_{2} = wτ^{ℓ_{2}}w^{−1}.

Applying the same analysis to σ_{1}, σ_{3} we may find u, δ such that

⟨uδu^{−1}⟩ ⊃ ⟨σ_{1}, σ_{3}⟩,

but δ ≠ vσv^{−1}, any v ≠ 1, δ ≠ σ^{k} any k > 1. Then we have σ_{1} = uδ^{k_{1}}u^{−1},

σ_{3} = uδ^{k_{3}}u^{−1},

some k_{1}, k_{3}. Thus in particular

τ^{ℓ_{1}}w^{−1} = uδ^{k_{1}}u^{−1},

and then the minimality assumptions on τ and δ give

w = u,
τ = δ,

and we are happy. (□Claim)

Thus we have established that the cocycle γ_{j}, derived from ρ̂ and projection to the j^{th} copy of the free group in H, is almost everywhere trivial for elements of G which have the identity on the i^{th} coordinate. In other words, if we let

Ĝ = 𝔽_{2} × 𝔽_{2} × ...{1}^{(i^{th}place)} × 𝔽_{2}... × 𝔽_{2} ◁ G,

and

Ĥ = 𝔽_{2} × 𝔽_{2} × ...{1}^{(j^{th}place)} × 𝔽_{2}... × 𝔽_{2} ◁ H,

then for all g⃗ ∈ Ĝ and almost all x ∈ X there will exist some h⃗ ∈ Ĥ with

h⃗ · ρ̂(x) = ρ̂(g⃗ · x).

In otherwords, ρ̂ is a measurable homomorphism of EX ; Ĝ to EY ; Ŷ, and we can appeal to the inductive assumption.

□

Standing back somewhat from the specifics of the proof one can abstract the following:

Theorem 3.4 Let G = G_{1} × G_{2} × ...G_{n+1} and H = H_{1} × ...H_{m} be countable groups where m ≤ n and each H_{i} is either the free group or abelian and G_{n+1} is amenable.

Suppose G acts by mpts on standard Borel probability space (X, μ), where the action of G_{n+1} is ergodic and the action of each G_{i}, i ≤ n is E_{0}-ergodic. Suppose H acts freely by Borel transformations on a standard Borel space Y.

Then E_{G}^{X} is E_{H}^{Y}-ergodic.

Corollary 3.5 For m ≤ n + 1 and S countable, the shift action of (𝔽_{2})^{n+1} on {0, 1}^{(𝔽_{2})^{n+1}×S)} is relatively ergodic with respect to any E_{Y}^{(𝔽_{2})^{m}}, where (𝔽_{2})^{m} acts freely by Borel transformations on standard Borel automorphisms.

Corollary 3.6 For m, n, Y as above, and

π : (𝔽_{2})^{n+1} ↠ (𝔽_{2})^{m+1}

the projection onto the first m + 1 coordinates, if we let (𝔽_{2})^{n+1} act on

X = {0, 1}^{(𝔽_{2})^{n+1}} × {0, 1}^{(𝔽_{2})^{m+1}}

by

g · (x_{1}, x_{2}) = (g · x_{1}, π(g) · x_{2}),

then E_{(𝔽_{2})^{n+1}}^{X} is E_{(𝔽_{2})^{m}}^{Y}-ergodic.

Proof Write (𝔽_{2})^{n+1} as (𝔽_{2})^{m+1} × S_{0}, where S_{0} = (𝔽_{2})^{m−n}. We take one more step in this direction and let S = S_{0} ∪ {∗}, where ∗ is any point not in S_{0}, and observe that, under its natural inclusion into (𝔽_{2})^{n+1}, the action of (F_{2})^{m+1} on X is isomorphic in a measure preserving fashion to the shift action of (𝔽_{2})^{m+1} on

{0, 1}^{(𝔽_{2})^{m+1}×S}.

□

Thus we obtain a funny kind of “invariant” which can be used to distinguish between different kinds of equivalence relations induced by the free measure preserving actions of some (𝔽_{2})^{ℓ} on a standard Borel probability space. We can consider the largest m < ℓ such that we obtain a relative ergodicity result with respect to free actions of (𝔽_{2})^{m}.

The example of the last corollary shows that this largest m can be any natural number less than ℓ.

4 Amenable classes of structures

In what follows we will adopt the convention that whenever we consider a class of structures C we have in mind that they are all on some fixed countable set. Most of the time that set could be taken to be ℕ, but it is notationally convenient to allow domains such as ℕ^{n}.

Definition We say that a countable Borel equivalence relation E on X is amenable if there is an E-invariant assignment

x ↦ φ_{x}

assigning a mean to each equivalence class [x]_{E} which has the property that whenever μ is a Borel probability measure on X and

f : E → [−1, 1]

is Borel then

x ↦ φ_{x}(f_{x})

is μ-measurable, where f_{x} : [x]_{E} → [−1, 1] is defined by

y ↦ f(x, y).

Definition We say that a standard Borel class of structures C is essentially countable if

≅ |_{C} ≤_{B} E_{∞};

that is to say, if its isomorphism relation is essentially countable.

Definition We say that a class of countable structures C, each having underlying set ℕ^{n}, some n ∈ ℕ, is amenable if there is an assignment

M ↦ φ_{M}

of means on ℕ^{n} to structures in C such that whenever π : M ≅ N is an isomorphism, then

φ_{N} ∘ π = φ_{M},

and moreover whenever μ is a standard Borel probability measure on X and f : C × ℕ^{n} → [−1, 1] is Borel then

M ↦ φ_{M}(f_{M})

is μ-measurable.

Example (Compare [7].) Assume the continuum hypothesis, and let C be the structures on ℕ which are isomorphic to

(ℤ, <, U),

the integers under the usual ordering, equipped with some unary predicate U ⊂ ℤ which will depend on the particular M ∈ C. For each M ∈ C we have may in a Borel manner choose an isomorphism

ψ : M ≅ ℤ,

and then at each n let φ_{M,n} : [−1, 1]^{M} → [−1, 1]

f ↦ \frac{1}{2n + 1} ∑_{i=−n}^{i=n} f(i).

Using Modobodzki’s limit medial we may find a universally measurable ultrafilter U on ℕ and for each M let

φ_{m} = ∫_{U} φ_{M,n} :: [−1, 1]^{M} → [−1, 1]

f ↦ ∫_{U} \frac{1}{2n + 1} ∑_{i=−n}^{i=n} f(i).

We have to face up to the fact that C may be amenable without being essentially countable: For instance consider the models of [13], consisting of expansions of (ℕ · ℤ, <^{lex}); as discussed in [3], this isomorphism relation is far from essentially countable, but if we assume CH then we can, in parallel to the example above, use Modobodzki to obtain an invariant and suitably measurable assignment of means. Worse: The countable class of structures may be amenable and essentially countable without ≅|_{C} being Borel reducible or universally measurably reducible to an amenable countable Borel equivalence relation. If we take any class of structures B whatsoever and replace it by C = {M ˙∪(ℤ, <) : M ∈ B} then under CH we may assign suitably invariant means by concentrating on the (ℤ, <), while maintaining ≅ |_{B} ≤_{B}≅ |_{C}.

This difficulty is the subject of some rather tortured lemmas below. Under suitable assumptions amenability, to the effect that the amenability is suitably ubiquitous, we do indeed obtain that essentially countability will entail reduction to an amenable countable Borel equivalence relation.

The second problem is that without CH there is no known way to obtain any examples of measure amenable structures.

This will ultimately be bypassed by a standard metamathematical trick. The statements we will be proving are all sufficiently absolute, and it turns out that we may simply impose the assumption that the continuum hypothesis holds.

Perhaps there is a more elegant solution, and a better way to formulate a relevant concept of amenability. The next lemma is implicit in [5].

Lemma 4.1 Let C be a class of structures with

θ : C → X_{∞}

witnessing ≅ |_{C} ≤_{B} E_{∞}. Then we may find a countable Borel equivalence relation F on standard Borel Y with γ : C → Y witnessing ≅ |_{C} ≤_{B} F and we may in a Borel manner assign to each M ∈ C a pair (A_{M}, ρ_{m}) such that

(a) A_{M} ⊂ ⋃_{n} M^{n};

(b) ρ_{M} : A_{M} → Y;

(c) M ↦ (A_{M}, ρ_{M}) is ≅-invariant, in the sense that if π : M ≅ N then

π[A_{M}] = A_{N} ,
ρ_{M} = ρ_{N} ∘ π;

(d) ρ_{M}(a⃗)F (M) all a⃗ ∈ A_{M};

(e) Y equals the range of γ;

(f) there is Borel ψ : Y → C such that ψ(γ(M)) ≅ M all M ∈ C.

Proof Following [5] we may find a countable fragment F ⊂ L_{ω_{1},ω} such that for each M ∈ C there will be some a⃗ ∈ M^{<}ℕ such that ⟨M; a⃗⟩ is F -atomic. The following are all routine consequences of the definition of atomicity:

(i) the set {⟨M, a⃗⟩ : ⟨M, a⃗⟩ is F -atomic } is Borel;

(ii) in a given fragment F_{n} = F (c_{1}, ..., c_{n}) (where c_{1}, ..., c_{n} are fresh constant symbols), the collection of T ⊂ F_{n} which are complete and admit an atomic model is Borel;

(iii) for each T as in (ii) we may in a Borel way choose some ⟨M, a⃗⟩ with ⟨M, a⃗⟩ ⊧ T. Given these facts, we let

A_{M} = {a⃗ : ⟨M, a⃗⟩ isF −atomic},
ρ_{M}(a⃗) = Th_{F} (⟨M, a⃗⟩),
Y = {Th_{F} (⟨M, a⃗⟩) : a⃗ ∈ A_{M}},

and for each T ∈ Y we choose M ∈ C as (iii). □

Definition For C a class of structures, we let C^{[n]} = {M^{n} : M ∈ C}, the class of all structures arising as n-fold products of a structure from C.

Lemma 4.2 If C is an amenable class of structures on ℕ then C^{[n]} is an amenable class of structures on ℕ^{n}.

Proof Let M ↦ φ_{M} be the invariant and universally measurable assignment of means. We then define φ_{M}^{ℓ} recursively on ℓ, beginning with

φ_{M}^{1} = φ_{M}

and continuing with

φ_{M}^{ℓ+1}(f) = φ_{M}(a ↦ φ_{M}^{ℓ}(f_{a})),

where f_{a} : M^{ℓ} → [−1, 1] is defined by f_{a}(b_{1}, b_{2}, ..., b_{ℓ}) = f(a, b_{1}, b_{2}, ..., b_{ℓ}). The invariance property for the various φ_{M}^{ℓ}’s is easily checked. The main concern is measurability.

Given any measure μ on C^{[n]}, for any a_{1}, ..., a_{n−1} ∈ ℕ we have

M ↦ φ_{M}(a_{n} ↦ f(a_{1}, a_{2}, ..., a_{n}))

μ-measurable, and hence we can find some μ-conull set A_{1} ⊂ C such these functions are all Borel on this set. Then the functions

A_{1} → [−1, 1]
M ↦ φ_{M}(a_{n−1} ↦ φ_{M}(a_{n} ↦ φ_{M}(f(a_{1}, a_{2}, ...a_{n−1}, a_{n})))

are μ-measurable, and we again can go to a conull A_{2} ⊂ A_{1} on which they are all Borel, and continue in this fashion until we reach A_{n}. □

Lemma 4.3 Let C be an essentially countable class of amenable structures on ℕ with the futher property that given any Borel and ≅-invariant assignment

C → P(ℕ^{n})
M ↦ A_{M}

we can choose an assignment of means

M ↦ φ_{M}

which not only witnesses amenability but has the further property of always concentrating on A_{M} – that is to say, φ_{M}(χ_{A_{M}}) is always 1.

Then ≅ |_{C} is Borel reducible to an amenable countable Borel equivalence relation.

Proof Following 4.1 we may choose M ↦ (A_{M}, ρ_{M}), γ, ψ, Y, F all as indicated there. At each n let Y_{n} = {y ∈ Y : for ⟨M, a⃗⟩ = γ(y), a⃗ ∈ M^{n}}; it suffices to show that each F |_{Y_{n}} is amenable.

But for this purpose we can choose an assignment of means

M ↦ φ_{M}

such that each φ_{M} concentrates on M^{n} ∩ A_{M}. (In the case that M^{n} ∩ A_{M} is empty, then M plays no part in Y_{n} and we could for instance ask that φ_{M} simply concentrate on M^{n}.)

Then for y ∈ Y_{n} we let

φ_{y} : [−1, 1]^{[y]_{F}} → [−1, 1]
f ↦ φ_{ψ}(a⃗ ↦ f(ρ_{M}(a⃗))).
□

Corollary 4.4 Let C be a class of amenable structures and suppose that for all M ∈ C and a ∈ M

M ⊧ ∀b ⋁_{t a term} t(a) = b;

that is to say, the algebraic closure of any a ∈ M is the whole structure.

Then ≅ |_{C} is Borel reducible to an amenable countable Borel equivalence relation.

Proof It is easily seen, and observed in the course of [5], that since structures in C are finitely generated we must have ≅ |_{C} essentially countable. Let (t_{i})_{i∈N} be some fixed enumeration of terms.

Given any invariant Borel assignment M ↦ A_{M} ⊂ M^{n} and invariant measurable

M ↦ φ_{M}

we can define for each M ∈ C

σ_{M} : M ↦ A_{M}

by

σ_{M}(a) = (t_{i(1)}(a), t_{i(2)}(a), ..., t_{i(n)}(a))

where the sequence i(1), i(2), ..., i(n) is chosen lexicographically least to ensure

(t_{i(1)}(a), t_{i(2)}(a), ..., t_{i(n)}(a)) ∈ A_{M}.
□

5 Non-reduction

Throughout this section we assume the continuum hypothesis. We do this without entertaining any philosophical bias either for or against CH, but simply because the statements we are attempting to prove are all Π ∼_{1}^{1} and easily absolute between the universe and some extension where we have forced a wellorder of 2^{ℵ_{0}} of ordertype ℵ_{1}.

We first need a technical result. This result can be most easily proved by collecting together the observations of the last section.

Lemma 5.1 Let G and H = H_{1} × H_{2} × ...H_{N} be countable groups acting by Borel transformations on standard Borel X and Y, with the H acting freely, and let

ρ : X → Y

be a homomorphism of E_{G}^{X} to E_{H}^{Y} with

β : X × G → H

the induced cocycle, and

β_{j} = π_{j} ∘ β : X × G → H_{j}

the induced cocycle for the various H_{j}.

Suppose that at each j ≤ N we have either (a) β_{j} maps into an abelian subgroup of H_{j}, or

(b) there is a hyperfinite equivalence relation F_{j} on some standard Borel Z_{j} and a homomorphism ρ̂: X → Z_{j} from E_{G}^{X} to F_{j} with the property that for all x ∈ X, g ∈ G

ρ̂_{j}(g · x) = ρ̂_{j}(x) ⇔ β_{j}(x, g) = 1.

Then there is an amenable countable Borel equivalence relation F̂ on Ẑ such that ρ factors through F̂.

Proof We may as well assume that N = 2, H_{1} is abelian, and that there is homomorphism

ρ̂: X → Z

from E_{G}^{X} to some hyperfinite F with the additional property for all x ∈ X, g ∈ G

ρ̂(g · x) = ρ̂(x) ⇔ β_{2}(x, g) = 1.

We make the further harmless assumption that each ρ̂[[x]_{G}] is infinite; the case for those which are finite can be dealt with separately using an argument which is similar to and somewhat easier than the one below.

Since hyperfinite equivalence relations are induced by ℤ-actions (see [6], [7]), we may in a Borel fashion assign

z ↦<_{z},

where each <_{z} is a linear order of ordertype ℤ on [z]_{F} with the the invariance property that

z_{1}F z_{2} ⇒<_{z_{1}}=<_{z_{2}} .

We then write

xR_{ℓ}x′

if ρ̂(x)F ρ̂(x′) and ρ̂(x), ρ̂(x′) are exactly ℓ many places apart in the linear order <_{ρ̂(x)}=<_{ρ̂(x′)} restricted to ρ̂[[x]_{G}]. In other words, if there are x_{0}, x_{1}, ..., x_{ℓ} ∈ [x]_{G} with

x_{0} = x,

x_{ℓ} = x′,
ρ̂(x_{0}) = ρ̂(x),
ρ̂(x_{ℓ}) = ρ̂(x′),

each

ρ̂(x_{i}) <_{ρ̂(x)} ρ̂(x_{i+1})

and for all x̂ ∈ [x]_{G} with

ρ̂(x_{0}) <_{ρ̂(x)} ρ̂(x̂) <_{ρ̂(x)} ρ̂(x_{ℓ})

there will be some i with

ρ̂(x̂) = ρ̂(x_{i}).

This definition makes sense for ℓ ≥ 0, and we can extend it in the natural way toℓ < 0 by setting

x′R_{ℓ}x

if xR_{−ℓ}x′.

We will build a class of amenable structures C and then a class of expansions D. We will show that ρ factors through ≅ |_{D} and then appeal to 4.4 to argue that ≅ |_{D} is bi-borel reducible to an amenable countable Borel equivalence relation.

The language for C will have unary functions (F_{ℓ})_{ℓ∈ℤ}, (F̂_{h})_{h∈H_{1}}, and an equivalence relation E^{∗}. The structures in C are those structures on ℕ that satisfy:

(i) ∀a([a]_{E^{∗}} = {F̂_{h}(a) : h ∈ H_{1}}); for all h_{1}, h_{2} ∈ H_{1}

F̂_{h_{1}} ∘ F̂_{h_{2}} = F̂_{h_{1}+h_{2}},

∀b(F̂_{0}(b) = b);
(ii) ∀a, b

⋁_{ℓ∈ℤ} ⋁_{h∈H_{1}}(F̂_{h} ∘ F_{ℓ}(a) = b);

(iii) for all ℓ ∈ ℤ, h_{1}, h_{2}, h_{3}, h_{4} ∈ H_{1}

∀a, b ⋁_{h̄∈H_{1}}(F̂h̄ ∘ F̂_{h_{1}} ∘ F_{ℓ} ∘ F̂_{h_{2}}(a) = F̂_{h_{3}} ∘ F_{ℓ} ∘ F̂_{h_{4}}(a));

(iv)

∀a ⋁_{h∈H_{1}}(F̂_{h}(a) = F_{0}(a));

(v) for all ℓ_{1}, ℓ_{2} ∈ ℤ

∀a ⋁_{h∈H_{1}}(F̂_{h} ∘ F_{ℓ_{1}+ℓ_{2}}(a) = F_{ℓ_{1}} ∘ F_{ℓ_{2}}(a)).

In essence (i) states that the {F̂_{h} : h ∈ H_{1}} act transitively on each E^{∗}-equivalence class, and that the action respects the group structure suggested by H_{1}. (ii) states in particular that the algebraic closure of any point is the whole structure. (iii)-(v) state that in essence that ℤ, via the assignment ℓ ↦ F_{ℓ}, acts on the structure’s collection of all E^{∗} equivalence classes.

We will in fact that this class of structures is not only amenable but “2-amenable” in the sense of [7]. We begin with a Folner sequence (D_{n})_{n∈ℕ} for H_{1}, that is to say a collection with the property that

D_{n} ⊂ D_{n+1},

H_{1} = ⋃_{n} D_{n},

and for all h ∈ H_{1}

\frac{|D_{n}hΔD_{n}|}{|D_{n}|} → 0.

For N ∈ ℕ, M ∈ C, a ∈ M, we first let

f_{n}^{a,M} = \frac{1}{|D_{n}|}χ_{\{F_{h}(a):h∈D_{n}\}};

that is to say, f_{n}^{a,M}(b) equals 1/D_{n} if there exists h ∈ D_{n} with M ⊧ F̂_{h}(a) = b, and equals zero otherwise. We iterate and define for each n, m

f_{n,m}^{a,M} : M → [0, 1]

f_{n,m}^{a,M} = \frac{1}{2n + 1}( ∑_{ℓ∈[−m,m]} f_{n}^{F_{ℓ}(a),m}).

At each M ∈ C and a, b ∈ M with aE^{∗}b, it follows from (i) and the properties of the Folner sequence that

lim_{n→∞}||f_{n}^{a,M} − f_{n}^{b,M}||_{ℓ_{1}} → 0,

and then by (ii) we have that for all a, b ∈ M

lim_{m→∞}(limsup_{n→∞}||f_{n,m}^{a,M} − f_{n,m}^{b,M}||_{ℓ_{1}} → 0.

We massage this into the form required by §2.5 [7] by recalling that each M ∈ C has ℕ = {1, 2, 3, ...} as its underlying set, and letting

f_{n,m}^{M} = f_{n,m}^{1,M},

and thereby obtain that C is 2-amenable.

As discussed in [9], it follows from CH and Modobodki’s theorem that C is amenable in our sense above. The class D arises by expanding the structures in C through the addition of fresh unary predicates,

(U_{n})_{n∈ℕ}. For this purpose let us fix countable Borel separating family (B_{n})_{n∈ℕ} for Y. To each M ∈ C and and function

σ : M → Y

we associate the structure M^{σ} which includes the language of C, on which it behaves just like M, but interprets the fresh unary predicates by obeying the dictate that

(N^{σ} ⊧ U_{n}(a)) ⇔ σ(a) ∈ B_{n}.

We let D be the collection {M^{σ}|M ∈ C, σ : M → Y }. These satisfy the assumptions of 4.4 and so it suffices to show that ρ factors through ≅ |_{D}.

For each x we define a pair n_{x}, σ_{x} : N_{x} → Y. We will not literally have that N_{x} is in C because its underlying set, although countable, will not equal ℕ. Nonetheless we can simply choose an isomorphic copy N_{x}^{∗} ∈ C and argue that

M_{x} = (N_{x}^{∗})^{σ_{x}^{∗}} ∈ D,

for some natural choice of σ_{x}^{∗}, is as required.

The underlying set of N_{x} will be the set of pairs (ℓ, y) such for some x̂ ∈ [x]_{G} we have (a) xR_{ℓ}x̂;

(b) yE_{H_{1}}ρ(x̂).

We set

(ℓ, y)E^{∗}(ℓ̄, ȳ)

if and only if ℓ = ℓ̄. We naturally enough define the F̂_{h} functions as suggested by the action of H_{1}, so that

F̂_{h}(ℓ, y) = (ℓ, h · y).

The F_{ℓ} functions require more care.

First fix an enumeration (h̄_{i})_{i∈ℕ} of H_{1} × H_{2}. Then let

F_{ℓ}((y, ℓ̄)) = (h̄_{i_{0}} · y, ℓ + ℓ̄)

where i_{0} is least such that for some x̂ ∈ [x]_{G} we have

xR_{ℓ+ℓ̄}x̂,
(h̄_{i_{0}} · y)E_{H_{1}}ρ(x̂).

We then, in the most obvious way possible, define

σ_{x} : N_{x} → Y
(ℓ, y) ↦ y.

In a Borel manner it is possible to choice for each x a bijection

π_{x} : ℕ ≅ {(ℓ, y) : ℓ ∈ ℤ, ∃x̂ ∈ [x]_{G}(yE_{H_{1}}ρx̂, xR_{ℓ}x̂)},

and we finish the definion by letting N_{x}^{∗} be the copy on ℕ of N_{x} provided by π_{x} and then setting

M_{x} = (N_{x}^{∗})^{σ_{x}∘π_{x}}.

Claim(I): If x̂_{1}, x̂_{2} ∈ [x]_{G} with

xR_{ℓ}x̂_{1},
xR_{ℓ}x̂_{2},

then

[ρ(x̂_{1})]_{H_{1}} = [ρ(x̂_{2})]_{H_{1}}.

Proof of claim: Fix g ∈ G with g · x̂_{1} = x̂_{2}.

It follows from the definition of R_{ℓ} that ρ̂(x̂_{1}) = ρ̂(x̂_{2}), and hence

β_{2}(x̂_{1}, g) = 1
∴ β(x̂_{1}, g) ∈ H_{1} × {1}
∴ [ρ(x̂_{1})]_{H_{1}} = [ρ(x̂_{2})]_{H_{2}}.
(□Claim)

Claim(II): For any x ∈ X we have N_{x}^{∗} in C. Proof of claim: Appealing to claim(I) we have

{y : (ℓ, y) ∈ N_{x}}

is an H_{1}-equivalence class for any ℓ ∈ Z. This quickly implies (i) in the definition of C. (ii) follows from the careful choice of the F_{ℓ} functions. For (iii)-(v) we observe that the E^{∗}-equivalence classes are arranged into a ℤ-chain by {xR_{ℓ}(·) : ℓ ∈ ℤ} and that the action of the F_{ℓ} functions respects that ordering. (□Claim)

Claim(III): M_{x_{1}} ≅ M_{x_{2}} ⇒ ρ(x_{1})E_{H}ρ(x_{2}). Proof of claim: Since the functions σ_{x_{1}} ∘ π_{x_{1}} and σ_{x_{2}} ∘ π_{x_{2}} must have a point in common. (□Claim)

Claim(IV): ρ(x_{1})E_{H}ρ(x_{2}) ⇒ M_{x_{1}} ≅ M_{x_{2}}.

Proof of claim: If x_{1}E_{G}x_{2} then we can find some ℓ̂ with

x_{1}Rℓ̂x_{2},

and hence for all ℓ

x_{1}R_{ℓ+ℓ̂}(·) = x_{2}R_{ℓ}(·).

We simply define an isomorphism

ψ : N_{x_{1}} ≅ N_{x_{2}}

by

ψ(ℓ, y) = (ℓ − ℓ̄, y).

This isomorphism clealy intertwines σ_{x_{1}} and σ_{x_{2}}. (□Claim)

Thus ρ factors through ≅ |_{D} and we finish by an appeal to 4.4.

□

Theorem 5.2 Let G = G_{1} × G_{2} × ...G_{N} × K and H = (𝔽_{2})^{M} × ... × K̄ be countable groups, K ≅ ℤ, K̄ abelian, M < N, G acting by mpts on standard Borel (X, μ) and H acting freely by measure preserving transformations on standard Borel Y, with the action of K on (X, μ) being ergodic.^{1} Let ρ : X → Y be a homomorphism of E_{G}^{X} to E_{H}^{Y}.

Then at some i ≤ N, ρ viewed as a homomorphism from E_{G_{i}}^{X} to E_{H}^{Y} factors through a hyperfinite equivalence relation μ-a.e.

Proof We use ρ to define induced cocyles

β_{j} : X × G → H_{j}
β_{i,j} : X × G_{i} → H_{j}
β_{N+1,j} : X × K → H_{j}

for i ≤ N, j ≤ M in the usual way.

Definition: We say that a coordinate j ≤ M is passive for G_{i} is either:

(a) β_{i,j} is cohomologous to a cocycle into an abelian subgroup of H_{j}; or

(b) there is a standard Borel Z_{j} and hyperfinite countable F_{j} on Z_{j} and a homomorphism

ρ̂_{j} : X → Z_{j}

from E_{G_{i}} to F_{j} such for all g ∈ G_{i} and μ-a.e. x ∈ X

ρ̂_{j}(g · x) = ρ̂_{j}(x) ⇔ β_{i,j}(x, g) = 1.

1Here and beyond we identify K with {1}×...×{1}×K◁G, we identify G_{i} with {1}×...×G_{i}×{1}×...{1}×{0}◁G, we identify K̄ with {1} × ... × {1} × K̄ ◁ H, and in keeping with the notation of 3.3 we let H_{j} = {1} × {1} × ...𝔽_{2}^{(j^{th}place)} × {1}... × {0} ◁ H.

Naturally we want to advance this terminology and say that a coordinate is active if it is not passive.

Claim(I) If j ≤ M is active for G_{i} then there is a Borel assignment of measures

X ↦ M_{1,3}^{+}(∂H_{j})
x ↦ μ_{x}

such that for all k ∈ K and μ-a.e. x ∈ X

μ_{k·x} = β_{N+1,j}(x, k) · μ_{x}.

Proof of claim: Following appendix B we may certainly find a Borel assignment of measures

X ↦ M_{1}^{+}(∂H_{j})
x ↦ μ_{x}

such that for all k ∈ K and μ-a.e. x ∈ X μ_{k·x} = β_{N+1,j}(x, k) · μ_{x}. The task is to do this so that the measures do not simply concentrate on two or fewer ends.

Suppose then that we can find no such suitably invariant assignment with μ_{x} ∈ M_{1,3}^{+}(∂H_{j}) almost everywhere. Following the proof of a parallel step in 3.3 we may conclude that

μ_{g·x} = β_{i,j}(x, g) · μ_{x}

all g ∈ G_{i} and μ-a.e. x ∈ X.

Following A.6 and the notation discussed there, we obtain a split in cases:

Case(a) Almost everywhere μ_{x} ∈ B_{2}, the set of ν ∈ M_{1}^{+}(∂H_{j}) with non-trivial stabilizer under the action of H_{j}.

Then from A.6 we have E_{H_{j}}^{B_{2}} smooth, and so by appealing to the ergodicity of the K action we can find a single ν_{0} such that

μ_{x}E_{H_{j}}^{B_{2}}ν_{0}

almost everywhere. Appealing to the selection theorem for Borel sets with countable sections, we may find a Borel function

γ : X → H_{j}

such that

γ(x) · μ_{x} = ν_{0}

almost everywhere.

Now consider the new cocycle

α_{i,j} : X × G_{i} → H_{j}
(x, g) ↦ γ(g · x)β_{i,j}(x, g)γ(x)^{−1}.

This is certainly cohomologous to β_{i,j} and we have that almost everywhere

α_{i,j}(x, g) · ν_{0} = γ(g · x)β_{i,j}(x, g)γ(x)^{−1} · ν_{0}
= γ(g · x)β_{i,j}(x, g) · μ_{x} = γ(g · x) · μ_{g·x} = ν_{0}

and hence

α_{i,j}(x, g) ∈ (H_{j})_{ν_{0}},

the stabilizer of ν_{0}.

From A.6 this stabilizer is abelian, and H_{j} is passive for G_{i}.

Case(b) On a positive measure subset of X the associated μ_{x} is not in B_{2}.

Then appealing to the ergodicity of the action of K we can obtain that almost everywhere μ_{x} has trivial stabilizer. Thus almost everywhere we have

μ_{g·x} = μ_{x}

if and only if

β_{i,j}(x, g) · μ_{x} = μ_{x}

if and only if

β_{i,j}(x, g) = 1.

Thus if we let Z_{j} = B_{0} ∪ B_{1}, and define

ρ̂_{j} : X → Z_{j}
x ↦ μ_{x}

almost everywhere, then by A.6 we obtain after all that H_{j} is passive for G_{i}. (□Claim)

Claim(II): If j ≤ M is active for some G_{i} then it is passive for every other G_{i′}. Proof of claim: This parallels a similar point from 3.3.

Appealing to the last claim and A.1 we may find a Borel

ŝ: X → H_{j}

such that for any k ∈ K we have almost everywhere

ŝ(k · x) = β_{N+1,j}(x, k) · ŝ(x),

and thus if we replace

β_{j} : X × G → H_{j}

by the cohomologous cocycle

α_{j} : X × G → H_{j}
(x, g) ↦ ŝ(g · x)^{−1}β_{j}(x, g)ŝ(x)

then α_{j} is trivial for K. Appealing to 3.2 we have a homomorphism

π : G → H_{j}

with π(g) = α_{j}(x, g) almost everywhere, and we finish as in the parallel part of the proof of 3.3. (□Claim)

Thus by the pigeonhole principle we obtain some i ≤ N such that no j ≤ M is active for G_{i}. From this it follows that the cocycle

β|_{X×G_{i}} : X × G_{i} → H

is cohomologous to some

β^{∗} : X × G_{i} → H

such that each of the projections

β_{j}^{∗} : X × G_{i} → H_{j}

satisfy either (a) or (b) from the preceding lemma.

Fix γ^{∗} : X → H such that

γ(g · x)β(x, g)γ(x)^{−1} = β^{∗}(x, g)

almost everywhere. If we define

ρ^{∗} : X → Y
x ↦ γ(x) · ρ(x)

then ρ^{∗} is a homomorphism from E_{G_{i}}^{X} to E_{H}^{Y} and β^{∗} is the induced cocycle.

Thus by the lemma we may find an amenable countable Borel equivalence F′ on standard Borel Z′ and homomorphisms

ρ_{0}^{∗} : X → Z′
ρ_{1}^{∗} : Z′ → Y

of E_{G_{i}}^{X} to F and of F to E_{H}^{Y} such that

ρ_{1}^{∗} ∘ ρ_{0}^{∗} = ρ^{∗}.

As in the proof of 3.1 we may find Ẑ ⊂ Z′ such that Z′|Ẑ is hyperfinite and almost everywhere ρ_{0}^{∗}(x) ∈ Ẑ. We then take one further step and let Z = Ẑ × H,

ρ_{0} : x ↦ (ρ_{0}^{∗}(x), γ(x)),
ρ_{1} : (z, h) ↦ h^{−1} · ρ_{1}^{∗}(z)

and set (z, h)F (z′, h′) if and only if zF′z′. It is easily checked that F is hyperfinite, and for almost every x we have

ρ_{1} ∘ ρ_{0}(x) = ρ_{1}(ρ_{0}^{∗}(x), γ(x))
= γ(x)^{−1} · ρ_{1}^{∗}(ρ_{0}^{∗}(x))) = γ(x)^{−1} · ρ^{∗}(x) = ρ(x),

as required.

□

Corollary 5.3 (After Gaboriau?) Let G = (𝔽_{2})^{N} × ℤ and H = (𝔽_{2})^{M} × ... × ℤ, M < N, G acting freely by mpts on standard Borel probability space (X, μ) and H acting freely by measure preserving transformations on standard Borel Y, with the action of ℤ on (X, μ) being ergodic. Then

E_{G}^{X} ≰_{B} E_{H}^{Y}.

Proof In light of the theorem and B.2, it suffices to show that if F is a countable Borel equivalence relation on a standard Borel space Z, and

ρ : Z → {0, 1}^{ℕ}

is a homomorphism of F to E_{0}, and ρ^{−1}(w) is countable all w ∈ 2^{ℕ}, then F is hyperfinite.

But this follows from the fact that among the countable Borel equivalence relations the class of hyperfinite equivalence relations are closed under downward under ≤_{B}-reducibility and inclusion; see §1 [7]. □

Corollary 5.4 (After Adams-Kechris) There are ≤_{B}-incomparable countable Borel equivalence relations.

Proof On the one side we consider the Bernoulli shift of G = (𝔽_{2})^{3} × ℤ on {0, 1}^{G} equipped with the usual product measure and let X be the free part.

On the other side we first being with

H_{1} = (𝔽_{2})^{2} × ℤ
H_{2} = ℤ × (𝔽_{2})^{2}

along with their corresponding Bernoulli shifts on {0, 1}^{H_{1}} and {0, 1}^{H_{2}}. We let H = (𝔽_{2})^{2} × ℤ × (𝔽_{2})^{2}. We let Y_{1}, Y_{2} be the respectively free parts of these Bernoulli shifts and take Y = Y_{1} × Y_{2}. We let H act on Y by

(h_{1}, ℓ, h_{2}) · (y_{1}, y_{2}) = ((h_{1}, ℓ) · y_{1}, (h_{2}, ℓ) · y_{2}).

Since the product of mixing actions of ℤ are mixing, we obtain that the induced action of ℤ on Y is mixing and therefore certainly ergodic, and hence by the last corollary

E_{H}^{Y} ≰ E_{G}^{X}.

For the converse, suppose

ρ : X → Y

is a homomorphism of E_{G}^{X} to E_{H}^{Y}. We may write ρ = (ρ_{1}, ρ_{2}) where

ρ_{1} : X → Y_{1}
ρ_{2} : X → Y_{2}

are homomorphisms of E_{G}^{X} ot E_{H}^{Y_{1}} and E_{H}^{Y_{2}} respectively.

Appealing to 3.3 we obtain X_{1} ⊂ X of measure one such that ρ_{1}|_{X_{1}} has countable image, and then making the same appeal once more we have X_{2} of measure one with ρ_{2}|_{X_{2}} also having countable image. Thus ρ[X_{2}] is a countable subset of Y and we are done.

□

A The boundary of the free group

We adopt the notational convention that the generators of 𝔽_{2} be a and b.

Definition We let ∂𝔽_{2} be the boundary of the free group, consisting of all infinite reduced words in a and b – that is to say, those words with no consecutive appearances of a and a^{−1} or b and b^{−1}. We let 𝔽_{2} act in the obvious way on this collection, by concatenation and then cancellation to obtain an infinite reduced word. Note then that ∂𝔽_{2} is a compact space on which the free group acts by homeomorphisms.

[∂𝔽_{2}]^{2} be the collection of all unordered pairs or singletons from 𝔽_{2}. This is given a Borel structure in the following way: First we identify ∂𝔽_{2} with {{e} : e ∈ ∂𝔽_{}} ⊂ [∂𝔽_{2}]^{2}. We then fix a Borel linear ordering, <, of ∂𝔽_{2} × ∂F_{2} and identify

[∂𝔽_{2}]^{2} \ 𝔽_{2}

with

{(e_{1}, e_{2}) ∈ ∂𝔽_{2} × ∂𝔽_{2} : e_{1} < e_{2}}.

We extend the action of 𝔽_{2} on ∂𝔽_{2} by letting

σ · {e, e′} = {σ · e, σ · e′}.

The elements of ∂𝔽_{2} are also sometimes called the ends of the free group.

Definition Let M_{1}^{+}(∂𝔽_{2}) be the space of probability measures on ∂𝔽_{2}. We let M_{1,3}^{+}(∂𝔽_{2}) be those probability measures which do not simply concentrate on two ends – that is to say, for all e′, e′′ ∈ ∂𝔽_{2} we have μ({e′, e′′}) < 1.

Note that if we are given three distinct ends e_{1}, e_{2}, e_{3} ∈ ∂𝔽_{2} then we may select a common central branch point in a way which is firstly invariant under permutations of the sequence (e_{1}, e_{2}, e_{3}) and which secondly, and more importantly, commutes with the action of 𝔽_{2}. For instance, if we are presented with the ends

e_{1} = (aba^{−1}bba...)
e_{2} = (aba^{−1}a^{−1}ba...)
e_{3} = (aba^{−1}a^{−1}bb...)

then we would be led to choose s_{0}(e_{1}, e_{2}, e_{3}) = aba^{−2}b as their central point. Then if we were to act on each of them with the group element σ = ba^{2}b^{−1}a^{−1} we obtain

f_{1} = σ · e_{1} = (babba...)
f_{2} = σ · e_{2} = (bba..)
f_{3} = σ · e_{3} = (bbb...)

and are then confronted with s_{0}(f_{1}, f_{2}, f_{3}) = b^{2} as their branching point, which is indeed equal to σs_{0}(e_{1}, e_{2}, e_{3}). For measures in M_{1,3}^{+}(∂𝔽_{2}) one can use integration to extend this technique to find a center of the measure:

Theorem A.1 (Lyons) There is a Borel function

s : M_{1,3}^{+}(∂𝔽_{2}) → 𝔽_{2}

such that for all μ ∈ M_{1,3}^{+}(∂𝔽_{2}), σ ∈ F_{2}

s(σ · μ) = σs(μ).

Proof See [12] or [11]. □

Lemma A.2 The orbit equivalence relation E_{𝔽_{2}}^{∂𝔽_{2}} given by the action of 𝔽_{2} on ∂𝔽_{2} is hyperfinite.

Proof First consider the tail equivalence relation F on ∂𝔽_{2}. So given two ends e, e′ we set

eF e′

if there is some single end f and two words w_{1}, w_{2} with

e = w_{1}^{⌢}f,
e′ = w_{2}^{⌢}f.

It is known (see [6], [7]) that these kinds of tail equivalence relations are always hyperfinite.

Then since E_{𝔽_{2}}^{∂𝔽_{2}} ⊂ F we obtain that it must be hyperfinite as well. □

Lemma A.3 The orbit equivalence relation E_{𝔽_{2}}^{[∂𝔽_{2}]^{2}} induced by the left concatenation of 𝔽_{2} on [∂𝔽_{2}]^{2} is hyperfinite.

Proof First view [∂𝔽_{2}]^{2} as a subset of ∂𝔽_{2}×∂𝔽_{2} in the way indicated at the start of this section and consider the orbit equivalence relation E_{𝔽_{2}}^{∂𝔽_{2}×∂𝔽_{2}} given by the diagonal action of 𝔽_{2} on this product space. If we think of ∂𝔽_{2} × ∂𝔽_{2} as being a subset of infinite strings taken from 𝔽_{2} × 𝔽_{2} then the argument from A.2 indeed shows E_{𝔽_{2}}^{∂𝔽_{2}×∂𝔽_{2}} to be hyperfinite.

We clearly therefore have that the restricted equivalence relation E_{𝔽_{2}}^{∂𝔽_{2}×∂𝔽_{2}}|_{[∂𝔽_{2}]^{2}} is hyperfinite. Now since

E_{𝔽_{2}}^{[∂𝔽_{2}]^{2}} has finite index over the hyperfinite equivalence relation E_{𝔽_{2}}^{∂𝔽_{2}×∂𝔽_{2}}|_{[∂𝔽_{2}]^{2}} we finish by [7]. □

Lemma A.4 For any e ∈ ∂𝔽_{2}, the stabilizer of e in ∂𝔽_{2}, (𝔽_{2})_{e}, is abelian.

Proof Note that if e has non-trivial stabilizer then it must be eventually periodic. That is to say, viewed as infinite word it will have the form

e = w^{⌢}σ^{⌢}σ^{⌢}σ...

for some σ, w ∈ 𝔽_{2}. Let us assume we have represented e in this way and taken σ to be as short a word as possible – which is basically to say that σ does not have the form τ^{⌢}τ^{⌢}...τ for any other τ ∈ 𝔽_{2}.

After possibly replacing e by w^{−1} · e and conjugating (𝔽_{2})_{e} by w, we may as well assume w is the empty word and that e actually has the form σ^{⌢}σ^{⌢}σ^{⌢}.... Then it is easily seen that (𝔽_{2})_{e} = {σ^{ℓ} : ℓ ∈ ℤ}. □

Lemma A.5 For any {e, e′} ∈ [∂𝔽_{2}]^{2}, the stabilizer, (𝔽_{2})_{{e,e′}}, of {e, e′} in 𝔽_{2} is abelian.

Proof Following the proof of A.4 we may assume

e = σ^{⌢}σ^{⌢}σ...

where σ ≠ u^{k} for any u ∈ 𝔽_{2}, k > 1. Observe in particular that any string of the form σ^{⌢}σ^{⌢}σ...σ is already a reduced word and in particular σ does not have the form

σ = vσ_{0}v^{−1}

for any non-trival v ∈ 𝔽_{2}.

For any τ ∈ (𝔽_{2})_{{e,e′}} we must either have τ · e = e′ and τ · e′ = e or τ · e = e and τ · e′ = e′. In either case we end up discovering that τ^{2} · e = e.

Thus by the proof of the last lemma we in fact have to have τ^{2} = σ^{m} some m. It then follows by the observations regarding σ observed above that τ^{2} and hence τ cannot have the form vτ_{0}v^{−1} for any non-trivial v ∈ 𝔽_{2}.

Thus τ^{⌢}τ is a reduced word and equal to τ^{2}. From this it follows that m is even and τ = σ^{\frac{m}{2}} . □

The somewhat convoluted formulation of the next lemma is intentional. The statement given helps organize the proof.

Lemma A.6 We may divide M_{1}^{+}(∂𝔽_{2}) into three invariant pieces, B_{0}, B_{1}, B_{2}, such that:

(a) the orbit equivalence relation of 𝔽_{2} on B_{0} is smooth (i.e. ≤_{B} id(ℝ)) and the action is free (i.e. each μ has trivial stabilizer);

(b) the orbit equivalence relation of 𝔽_{2} on B_{1} is hyperfinite and the action is free;

(c) the orbit equivalence relation of 𝔽_{2} on B_{2} is smooth and each μ ∈ B_{2} has abelian stabilizer.

Proof We begin with B_{0} = M_{1,3}^{+}(∂𝔽_{2}). A.1 certainly implies that the action on B_{0} is free, and we see that the equivalence relation is smooth by taking

A_{0} = {μ ∈ M_{1,3}^{+}(∂𝔽_{2}) : s(μ) = 1}

as our selector.

We move on and let C be the set of linear combinations of the form

μ = α_{1}δ_{e_{1}} + α_{2}δ_{e_{2}}

where e_{1}, e_{2} ∈ ∂𝔽_{2}, δ_{e_{i}} is the Dirac point mass measure on e_{i}, α_{1} + α_{2} = 1, and α_{1}, α_{2} ≥ 0.We subdivide C into (C^{r})_{r∈[1/2,1]}, where α_{1}δ_{e_{1}} + α_{2}δ_{e_{2}} ∈ C^{r} if max{α_{1}, α_{2}} = r.

We then let B_{1}^{r} = {α_{1}δ_{e_{1}} +α_{2}δ_{e_{2}} ∈ C^{r}: one of e_{1}, e_{2} has non-trivial stabilizer in the action of 𝔽_{2} on ∂𝔽_{2}}. We let B_{1} be the union of these B_{1}^{r}’s. It follows from the argument of A.5 that the action of 𝔽_{2} on each B_{1}^{r} is free. We are left with trying to show that the orbit equivalence relation on such a B_{1}^{r} is hyperfinite.

In the case that r = \frac{1}{2} this directly reduces to A.3; in the case that r > \frac{1}{2} we have an invariant way to select from the pair {e_{1}, e_{2}} by chosing the end with highest weight, and it reduces to A.2.

Finally we take B_{2}^{r}{α_{1}δ_{e_{1}} + α_{2}δ_{e_{2}} ∈ C^{r}: both e_{1}, e_{2} have non-trivial stabilizer in the action of 𝔽_{2} on ∂𝔽_{2}}; we are then forced to let B_{2} be the union of these B_{2}^{r}’s. Note that the analysis of A.4 shows that there are only countably many possibilities for e_{1} and e_{2}; and so each B_{2}^{r} is the union of countably many orbits, and hence almost trivially smooth. □

In fact we can sharpen A.6 by observing that the stabilizers will be isomorphic to ℤ when non-trivial.

B Skew invariant measures

Lemma B.1 Let T be a measure preserving transformation of a standard Borel probability space (X, μ) and let ℤ act on X by

k · x = T^{k}(x).

Let α : X × ℤ → 𝔽_{2} be a cocycle.

Then we may find a measurable assignment

X → M_{1}^{+}(∂𝔽_{2})
x ↦ μ_{x}

such that for all k ∈ ℤ and almost all x ∈ X

μ_{k·x} = α(x, k) · μ_{x}.

Proof (Sketch only; chapter 4 of [17] contains much more general results.)

Let ℓ_{1}^{∞}(X, M_{1}^{+}(∂𝔽_{2})) be the space of all measurable assignments

γ : X → M_{1}^{+}(∂𝔽_{2}).

We equip this space with the topology provided by the semi-norms of the form

γ ↦ ∫_{X}(x ↦ ∫_{∂𝔽_{2}} f_{x}dγ(x))dμ

where x ↦ f_{x} is some measurable function from X to the functions in C(∂𝔽_{2}) of norm ≤ 1. In this topology ℓ_{1}^{∞}(X, M_{1}^{+}(∂𝔽_{2})) becomes a compact convex set. We let ℤ act on ℓ_{1}^{∞}(X, M_{1}^{+}(∂𝔽_{2})) by

(k · γ)(x) = α(−k · x, k) · γ(−k · x).

If we let

T_{N} : ℓ_{1}^{∞}(X, M_{1}^{+}(∂𝔽_{2})) → ℓ_{1}^{∞}(X, M_{1}^{+}(∂𝔽_{2}))

γ ↦ \frac{1}{N + 1} ∑_{k≤N} k · γ

and take any point in the intersection of the closures of the images of the operators T_{N} then we have a fixed point as required. □

This lemma can be used to give a neat proof that free actions of 𝔽_{2} on probability spaces are never hyperfinite, and that neat proof can in turn be used to show that the above lemma cannot be proved in the Borel, or even Baire measurable, category.

Corollary B.2 If 𝔽_{2} acts freely by mpts on standard Borel probability space (X, μ), then E_{𝔽_{2}}^{X} is not hyperfinite.

Proof Appealing to the ergodic decomposition theorem, we may assume without loss of generality that 𝔽_{2} acts ergodically. Write 𝔽_{2} = ⟨a, b⟩, so that a and b are the generators. We suppose for a contradiction that there is a Borel action of ℤ on X with E_{ℤ}^{X} = E_{𝔽_{2}}^{X} off of a null set.

We have a Borel cocycle

α : X × 𝔽_{2} → ℤ

given by

α(x, σ) = k

if k · x = σ · x. Appealing the last lemma we may find a measure one set X_{0} ⊂ X and a Borel assignment

x ↦ μ_{x}
X_{0} → M_{1}^{+}(∂𝔽_{2})

such that

μ_{ℓ·x} = α(x, ℓ) · μ_{x}

all x ∈ X_{0}, ℓ ∈ ℤ. Observing that for each x there will be ℓ_{a}^{x}, ℓ_{b}^{x} with

α(x, ℓ_{a}^{x}) = a,
α(x, ℓ_{b}^{x}) = b,

we obtain

μ_{a·x} = a · μ_{x},
μ_{b·x} = b · μ_{x},

all x ∈ X_{0}.

Claim: μ_{x} ∉ ∂M_{1,3}^{+}(∂𝔽_{2}) a.e. x.

Proof of claim: Otherwise ergodicity gives almost every μ_{x} ∈ M_{1,3}^{+}(∂𝔽_{2}) and appealing to A.1 we may find a Borel function

s^{∗} : X_{0} → 𝔽_{2}

such that for all σ ∈ 𝔽_{2} and a.e. x

s^{∗}(σ · x) = σ · s^{∗}(x).

σ_{0} ∈ 𝔽_{2} with Letting A_{σ} = {x ∈ X : s^{∗}(x) = σ} we have

(A_{σ})_{σ∈𝔽_{2}}

partitioning a conull subset of X into ℵ_{0} many disjoint sets of equal measure, with a contradiction to μ(X) = 1. (□Proof of claim)

So we actually may assume there is a Borel function

X_{0} → [∂𝔽_{2}]^{2}

x ↦ {e_{x}, e′_{x}}

such that

a · {e_{x}, e′_{x}} = {e_{a·x}, e′_{a·x}},
b · {e_{x}, e′_{x}} = {e_{b·x}, e′_{b·x}},

almost every x.

Let A_{0} be the set of x ∈ X_{0} such that neither e_{x} nor e′_{x} begin with a or a^{−1}. For each x ∈ X_{0} we may find some n ∈ ℕ with

b^{n} · x ∈ A_{0}.

Thus μ(A_{0}) = 0.

But considering that for n_{1} ≠ n_{2} ∈ ℕ

a^{n_{1}} · A_{0} ∩ a^{n_{2}} · A_{0} = ∅

we have our contradiction. □

Counterexample B.3 There is a Borel action of ℤ on a Polish space X and a Borel cocycle α : X ×ℤ → 𝔽_{2} such that there is no comeager X_{0} and Borel assignment

X_{0} → M_{1}^{+}(∂𝔽_{2})
x ↦ μ_{x}

with for all x ∈ X_{0}, n ∈ ℤ

μ_{n·x} = α(x, n) · μ_{x}.

Proof One constructs the example in this way. Again letting a and b be the generators of 𝔽_{2} we start with the free part of the shift action of 𝔽_{2} on {0, 1}^{𝔽_{2}}; notice that this action is generically ergodic, in the sense that every invariant Borel set is either meager or comeager. Following [15] we may find a comeager subset X of this free part on which

(a) E_{𝔽_{2}}|_{X} is hyperfinite; after some possible further contraction of X we may additionally suppose

(b) X is 𝔽_{2}-invariant;

(c) {a^{ℓ} · x : ℓ ∈ Z} is dense in X for all x ∈ X;

(d) {b^{ℓ} · x : ℓ ∈ Z} is dense in X for all x ∈ X. As indicated by say §3.1[4], it follows easily from (c) and (d) that the orbit equivalence relations E_{⟨a⟩}^{X}

and E_{⟨b⟩}^{X}, induced by the respective cyclic subgroups ⟨a⟩ and ⟨b⟩, are properly generically ergodic, and hence non-smooth, and in fact non-smooth even when restricted to any comeager subset of X. From (a) we get a Borel action of ℤ with

E_{ℤ}^{X} = E_{𝔽_{2}}^{X}.

Now for a contradiction suppose that on a comeager subset X_{1} ⊂ X we can find an assignment

x ↦ μ_{x}

which is suitably skew invariant under the cocyle from ℤ to 𝔽_{2}. As in the corollary above we would obtain

μ_{a·x} = a · μ_{x},
μ_{b·x} = b · μ_{x}

through out X_{1}. We may further assume that X_{1} is invariant under the action of 𝔽_{2}.

Claim: μ_{x} ∉ ∂M_{1,3}^{+}(∂𝔽_{2}) for a comeager set of x ∈ X_{1}.

Proof of claim: Otherwise generic ergodicity gives almost every μ_{x} ∈ M_{1,3}^{+}(∂𝔽_{2}) and, as before, we may find a Borel function

s^{∗} : X_{1} → 𝔽_{2}

such that for all σ ∈ 𝔽_{2} and a comeager set of X

s^{∗}(σ · x) = σ · s^{∗}(x).

Letting A_{1} = {x ∈ X : s^{∗}(x) = 1} we have that A_{1} meets all the orbits on a comeager subset X_{2} of X_{1} in exactly one point. Then we may find appeal to the uniformization theorem for Borel sets with countable sections to see that the function

f : X_{2} → A_{1}

which assigns to each x ∈ X_{2} the unique f(x)E_{𝔽_{2}}x with f(x) ∈ A_{1} will be 𝔽_{2} invariant, and hence constant on a comeager subset by generic ergodicity, with a contradiction to every orbit being meager. (□Proof of claim)

So as before may on a comeager subset X′ of X_{1} assign a pair of ends {e_{x}, e′_{x}} with

{e_{σ·x}, e′_{σ·x}} = σ · {e_{x}, e′_{x}}

all σ ∈ 𝔽_{2}, x ∈ X′. We can again let A_{0} be the set of x for which neither e_{x} nor e′_{x} begins with an a or an a^{−1}. By considering the action of ⟨b⟩ we analogously obtain that it will be non-meager. Thus by generic ergodicity of the action of ⟨a⟩ we obtain that for a comeager collection of x there will be some ℓ ∈ ℤ with a^{ℓ} · x ∈ A_{0}. Considering the definition of A_{0} and the invariance assumption on the {e_{x}, e′_{x}}_{x∈X′} we obtain that there can be at most one ℓ ∈ Z with a^{ℓ} · x ∈ A_{0}, and thus A_{0} is a Borel selector for a comeager subset of X, contradiction. □

C E_{0}-ergodicity

Lemma C.1 (Folklore; see [7].) A countable equivalence relation E is hyperfinite if and only if E ≤_{B} E_{0}.

Theorem C.2 (Jones, Schmidt; see [8]) Let G be a countable group acting ergodically by mpts on a standard Borel probability space (X, μ). Then E_{G}^{X} is E_{0} ergodic if and only if there is no sequence of sets (A_{n})_{n∈ℕ}, with each

μ(A_{k}) = \frac{1}{2},

and for all g ∈ G

μ(g · A_{k}ΔA_{k}) → 0

as k → ∞.

Without going into the depths of the proof, we may first tackle the if direction by supposing θ : X → {0, 1}^{ℕ} is a homomorphism of E_{G}^{X} to E_{0}. If [θ(x)]_{E_{0}} is not almost everywhere constant, then ergodicity implies that μ(θ^{−1}[{p}]), the measure of the pullback of the point p, is null for each p ∈ {0, 1}^{ℕ}. Then it is not hard to show that for each k there exists a set B_{k} ⊂ {0, 1} such that μ(θ^{−1}[B_{k}]) always equals a half and that for each f, g ∈ {0, 1}^{ℕ} with f(n) = g(n) all n > k we have

f ∈ B_{k} ⇔ g ∈ B_{k}.

Then A_{k} = θ^{−1}[B_{k}] gives us the required sequence.

For the converse, suppose that (A_{k}) is our asympotically invariant sequence of sets of measure one half. After possibly refining the sequence we may assume that for all g ∈ G there is a measure one set of x for which

∃N∀k > N(x ∈ A_{k} ⇔ g · x ∈ A_{k}).

Then we devise a measurable homomorphism by assigning to x ∈ X

f_{x} : ℕ → {0, 1}

with

f_{x}(k) = 1 ⇔ x ∈ A_{k}.

Theorem C.3 (Schmidt; see [14]) Let S be a countable set and equip {0, 1}^{𝔽_{2}×S} with the product measure μ. Then the shift action of 𝔽_{2} is E_{0} ergodic.

This result, which actually extends to any non-amenable countable group in the place of 𝔽_{2}, can be proved in various ways. It suffices to show that we can write the induced representation of 𝔽_{2} on L^{2}({0, 1}^{𝔽_{2}×S}, μ) as a product of the trivial representation and ℵ_{0} many copies of the regular left representation,

L^{2}({0, 1}^{𝔽_{2}×S}, μ) ≅ 1 ⊗ ⨂_{ℵ_{0}} ℓ^{2}(𝔽_{2}).

For this it suffices to show that the square integrable functions orthogonal to the constant function, that is to say those f ∈ L^{2}({0, 1}^{𝔽_{2}×S}, μ) with

∫ fμ = 0,

admit an orthonormal basis which is set wise invariant under the action of 𝔽_{2}.

Towards this goal we define for each finite F ⊂ 𝔽_{2} × S the function

A_{F} : {0, 1}^{𝔽_{2}×S} → {1, −1}

by

A_{F} (x) = −1

if the cardinality of {(σ, s) ∈ F : x(σ, s) = 1} is odd, and

A_{F} (x) = 1

if the cardinality is even.

Obviously this collection is set wise invariant. The Stone-Weierstrass theorem immediately implies that they span the functions orthogonal to the constant functions. We omit the routine calculation that they are orthogonal.

Things would certainly be nicer if mixing actions of 𝔽_{2} were necessarily E_{0}-ergodic. However it is known from say [16] that this is untrue. We give an elementary counterexample.

Counterexample We first consider the shift action of 𝔽_{2} on {0, 1}^{𝔽_{2}} and define a sequence of measures, (μ_{n})_{n∈ℕ}. We will make sure that these measures are 𝔽_{2}-invariant and that they are becoming increasingly “sticky”, with sets of measure a half which are nearly invariant for ever larger subset of 𝔽_{2}. We finish with the diagonal action of 𝔽_{2} on

∏_{ℕ}({0, 1}^{𝔽_{2}×S}, μ_{n}).

It suffices to define the measure μ_{n} on sets of the form

C_{s} = {x ∈ {0, 1}^{𝔽_{2}} : x|_{T} = s}

for s : T → {0, 1} and T ⊂ 𝔽_{2} a finite set which forms a tree under the usual Cayley graph for 𝔽_{2}.

For such an s and T we let (e_{i})_{i=1}^{i=|T |−1} enumerate the edges in T. We let K_{0} be the number of edges for which s takes the same value at both vertices at either end and we let K_{1} be the number of edges for which s assumes differing values at the two ends. We set

μ_{n}(C_{s}) = \frac{1}{2}(\frac{n − 1}{n})^{K_{0}}(\frac{1}{n})^{K_{1}}.

We omit the routine verification that this will indeed define a measure, and simply observe that for each g ∈ 𝔽_{2}

μ_{n}({x ∈ {0, 1}^{𝔽_{2}}}Δg · {x ∈ {0, 1}^{𝔽_{2}}})

gravitates towards zero as n marches towards infinity. Thus by C.2 the diagonal action of 𝔽_{2} on the product space ∏_{ℕ}({0, 1}^{𝔽_{2}×S}, μ_{n}) fails to be E_{0}-ergodic. Since each individual action of 𝔽_{2} on some individual ({0, 1}^{𝔽_{2}×S}, μ_{n}) is mixing, so too the overall diagonal action of 𝔽_{2} on the infinite product will be mixing.

References

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[16] Valette (?) recent book available on the internet [?] [17] R.J. Zimmer, Ergodic theory and semi-simple lie groups, Birkhauser,

greg@math.ucla.edu

MSB 6363 UCLA CA 90095-1555