Bi-Borel reducibility of essentially countable Borel equivalence relations^{∗}

Greg Hjorth^{†}

April 4, 2005

This note answers a questions from [3] by showing that considered up to Borel reducibility, there are more essentially countable Borel equivalence relations than countable Borel equivalence relations. Namely:

Theorem 0.1 There is an essentially countable Borel equivalence relation E such that for no countable Borel equivalence relation F (on a standard Borel space) do we have

E ∼_{B} F.

The proof of the result is short. It does however require an extensive rear guard campaign to extract from the techniques of [1] the following

Messy Fact 0.2 There are countable Borel equivalence relations (E_{x})_{x∈2^{ℕ}} such that:- (i) each E_{x} is defined on a standard Borel probability space (X_{x}, μ_{x}); each E_{x} is μ_{x}-invariant and μ_{x}-ergodic; (ii) for x_{1} ≠ x_{2} and A μ_{x_{1}}-conull, we have E_{x_{1}}|_{A} not Borel reducible to E_{x_{2}}; (iii) if f : X_{x} → X_{x} is a measurable reduction of E_{x} to itself, then μ_{x}(im(f)) > 0; (iv)

⋃_{x∈2^{ℕ}}{x} × X_{x}

is a standard Borel space on which the projection function

(x, z) ↦ x

is Borel and the equivalence relation Ê given by

(x, z)Ê(x′, z′)

if and only if x = x′ and zE_{x}z′ is Borel; (v)

2^{ℕ} → M( ⋃_{x∈2^{ℕ}}{x} × X_{x})

x ↦ δ_{x} × μ_{x}

is Borel.

∗Key words and phrases: Equivalence relation, Borel reducibility, near hyperbolic group, rigidity. 2000 Mathematics Subject Classifications: Primary 03E15. Secondary 20F67, 37A20.

†Partially supported by NSF grant DMS 0140503

We first prove the theorem granted this messy fact. We then prove the fact.

(iv) and (v) are messy and unpleasant to state precisely, but are intended to express the idea that we have an effective parameterization of countable Borel equivalence relations by points in a standard Borel space. Examples along these lines appear already in the Adams-Kechris constructions; the new feature is (iii).

Simon Thomas has pointed out to me that in light of theorem 4.4 [5] the Gefter-Golodets examples of section 5 [5] also satisfy the conclusion of 0.2. None the less I think it is worth including the construction below, since it ultimately relies on the far more elementary techniques of [1].

Acknowledgments: I am exceedingly grateful to an anonymous referee for a thorough and critical reading of the original version of this paper and for making many helpful comments.

1 Introduction

We include a few of the main definitions, some notational conventions, and facts about measure theory in the context of descriptive set theory. A more complete introduction to the theory of Borel equivalence relations can be found in [1], [3], or [5].

Definition For equivalence relations E, F defined on Polish spaces X, Y we write

E ≤_{B} F

if there is a Borel function f : X → Y with

x_{1}Ex_{2} ⇔ f(x_{1})F f(x_{2}).

We write E ∼_{B} F if E ≤_{B} F and F ≤_{B} E. We say that E is countable if every equivalence class is countable and Borel if it is Borel as a subset of X ×X. We say that it is essentially countable if there is some countable, Borel F with E ≤_{B} F.

Definition For X a Polish space we denote by M(X) the collection of Borel probability measures on X. This is given the σ-algebra generated by sets of the form

{μ : a < ∫ fdμ < b}

for f a Borel function on X and a, b ∈ ℝ. Following §17[2] M(X) becomes a standard Borel space. For x ∈ X we use δ_{x} to denote the Dirac measure with δ_{x}({x}) = 1 and δ_{x}(X \ {x}) = 0. For φ(·) some formula, possibly with parameters, and μ ∈ M(X) we write

∀^{μ}x(φ(x))

if the set of x ∈ X with φ(x) is co-null and

∃^{μ}x(φ(x))

if the set of x ∈ X with φ(x) is non-null.

Proposition 1.1 For Y Polish and A ⊂ X × Y Borel, the set

{(y, μ) ∈ Y × M(X) : ∃^{μ}x ∈ X((x, y) ∈ A)}

is Borel.

Proof It suffices to see that for a > 0 the sets

{(y, μ) ∈ Y × M(X) : μ({x : (x, y) ∈ A})} > a
{(y, μ) ∈ Y × M(X) : μ({x : (x, y) ∉ A})} > a

are both Σ ∼_{1}^{1}, which follows from 29.26 [2]. □

Definition For a group G acting on a set X and F ⊂ X we use stab_{F} to denote the set wise stabilizer – that is to say, the collection of g ∈ G such that {g · x : x ∈ F } = F. Thus in particular, for x ∈ X, stab_{{x}} denotes the group elements leaving x fixed.

In the case that G acts on a set X we will use [X]^{n} to denote the size n subsets of X equipped with the natural action

g · {x_{1}, ..., x_{n}} = {g · x_{1}, ..., g · x_{n}}.

We use [X]^{<∞} to denote the union of the [X]^{n}’s.

Lemma 1.2 The action of G on its finite subsets, [G]^{<∞}, is free if and only if G is torsion free.

Proof If g has infinite order, then for any h ∈ G, {g^{n} · h : n ∈ ℕ} is infinite, and thus there is no g-invariant finite subset of G. Conversely, if H is a finite subgroup of G then it is a fixed point in [G]^{<∞} for any h ∈ H. □

Lemma 1.3 If G acts freely on X, then there is a G-map from X to G.

Proof There is no requirement that this G-map be Borel, and thus we can select from each orbit [x]_{G} a unique point x_{0} ∈ [x]_{G}. From this we obtain a well defined map by letting θ(g · x_{0}) = g. □

We will at some later point need the following corollary.

Corollary 1.4 If G is a torsion free group then there is a G-map from [G]^{<∞} to G.

Definition For X a standard Borel space, we let M_{≤n}(X) be the probability measures which concentrate on at most n points and M_{≥n}(M) the measures which do not concentrate on at most n − 1 points. Thus M_{≥n}(X) = M(X) \ M_{≤n−1}(X).

The next lemma is a routine generalization of the argument given at C2.3 [1] along with credit to Lyons.

Lemma 1.5 Suppose that G is a group, X is a standard Borel G-space, Y is a countable G-space, and there is a Borel G-map from [X]^{n} to Y. Then there is a Borel G-map from M_{≥n}(X) to [Y]^{<∞}.

Proof Let ϕ be the Borel G-map from [X]^{n} to Y. Let {∗} be some single point not in Y and form the space Y^{∗} = Y ˙⋃{∗}. Define the Borel map

η : X^{n} → Y^{∗}

by η(x_{1}, x_{2}, ..., x_{n}) = ϕ({x_{1}, x_{2}, ..., x_{n}}) if the points are distinct, and equal to the default value ∗ otherwise. For μ ∈ M_{≥n}(X) we let μ^{∗} be the measure on Y^{∗} which has at each y_{0} ∈ Y^{∗}

μ^{∗}({y_{0}}) = μ^{n}({(x_{1}, ...x_{n}) : η(x_{1}, ...x_{n}) = y_{0}}).

The assumption that μ ∈ M_{≥n}(X) gives that μ^{∗} does not concentrate entirely on the new point ∗ of Y^{∗}, and thus we can select the finitely many points in Y for which μ^{∗}({y_{0}}) realizes the maximum value.

It is routinely seen that this assignment is Borel. □

2 Proof of theorem granted messy fact

We begin with a collection of (E_{x}, X_{x}, μ_{x}) as described there. Let A ⊂ 2^{ℕ} be Σ ∼_{1}^{1} non-Borel. Let C ⊂ ℕ^{ℕ} ×2^{ℕ} be closed with the canonical projection function

p_{2} : ℕ^{ℕ} × 2^{ℕ} → 2^{ℕ}

(z, x) ↦ x

having

p_{2}[C] = A;

so C is our witness to A ∈ Σ ∼_{1}^{1}.

We let X̂ be the set of (z, x, w) which have (z, x) ∈ C and w ∈ X_{x}. We give X̂ the standard Borel structure it inherits as a Borel subset of

ℕ^{ℕ} × ⋃_{x∈2^{ℕ}}{x} × X_{x}.

For v⃗_{1} = (z_{1}, x_{1}, w_{1}), v⃗_{2} = (z_{2}, x_{2}, w_{2}) ∈ X̂, we set

v⃗_{1}Ev⃗_{2}

if and only if x_{1} = x_{2} and w_{1}E_{x_{1}}w_{2}. This is clearly an essentially countable equivalence relation.

For a contradiction suppose F is a countable Borel equivalence relation on a standard Borel space Y and θ : X̂ → Y

ρ : Y → X̂

witness

E ∼_{B} F.

Let ρ̄: Y → ⋃_{x∈2^{ℕ}}{x} × X_{x} be obtained by setting ρ̄(y) = (p_{2}(ρ(y)), p_{3}(ρ(y))) – so if ρ(y) = (z, x, w) then ρ̄(y) = (x, w). Claim 2.1 If w_{0} ∈ X_{x} then {y : ρ̄(y) = (x, w_{0})} is countable. Proof For any y_{0} with ρ̄(y_{0}) = (x, w_{0}) we have that for all other y with

ρ̄(y) = (x, w_{0})

that yF y_{0}; thus the claim follows from F being countable. □ Claim 2.2 If x ∈ A then

{x} × X_{x} ∩ im(ρ̄)

is non-null in

({x} × X_{x}, δ_{x} × μ_{x}).

Proof We let z be such that (z, x) ∈ C. We define

γ_{z,x} : X_{x} → Y

by γ_{z,x}(w) = θ(z, x, w); the assumptions on θ entail that γ_{z,x} witnesses E_{x} ≤_{B} F. By composition we have

ρ ∘ γ_{z,x} : X_{x} → X̂

witnessing E_{x} ≤_{B} E. And then by μ_{x}-ergodicity of E_{x} we have some single x_{0} such that

∀^{μ_{x}}w ∈ X_{x}(ρ̄(γ_{z,x}(w)) ∈ {x_{0}} × X_{x_{0}}).

Postcomposing with the projection to the second coordinate we obtain a reduction

γ̄_{z,x} = p_{2} ∘ ρ ∘ γ_{z,x}
X_{x} → X_{x_{0}}

on a conull subset of E_{x} to E_{x_{0}}, and hence x = x_{0} by part (ii) of our messy fact. And then by part (iii) μ_{x}(im(γ̄_{z,x})) > 0, and, after unraveling the definitions, im(ρ̄) non-null in {x} × X_{x}. □

Putting together these claims we have

\begin{aligned}x ∈ A & ⇔ ∃^{μ_{x}}w ∈ X_{x}((x, w) ∈ im(ρ̄)) \\ & ⇔ ∃^{δ_{x}×μ_{x}}(x′, w′) ∈ ⋃_{y∈2^{ℕ}}\{y\} × X_{y}((x′, w′) ∈ im(ρ̄)) \end{aligned} .

ρ̄ is countable to one, and so it has Borel image by Lusin-Novikov (see 18.10 [2]). Applying lemma 1.1 we have A is Borel with a contradiction.

3 Messy fact

Notation For p⃗ = (p_{1}, p_{2}, ...) a sequence of distinct primes, let ℚp⃗ be the subgroup of ℚ generated by

{(p_{i})^{−n} : i, n ∈ ℕ}

(so that we have infinite divisibility in ℚp⃗ by each prime on the list). Then let

Γp⃗ = Hp⃗ ∗ Gp⃗

where Hp⃗ and Gp⃗ are disjoint copies of ℚp⃗.

Following §C3 of [1] we let [S_{Γp⃗}] be the collection of all infinite sequences

(a_{0}, a_{1}, a_{2}, ...a_{n}, ...)

such that at each i we have a_{i} ∈ Hp⃗ iff a_{i+1} ∈ Gp⃗ iff a_{i} ∉ Gp⃗; we let Γp⃗ act in the natural way on [S_{Γp⃗}] – concatenate and reduce.

Lemma 3.1 (a) If w, a ∈ Γp⃗ with g^{2} = w^{⌢}a^{⌢}w^{−1} then g is in the cyclic subgroup generated by wcw^{−1} for any c with c^{2} = a.

(b) If a non-identity element h of Γp⃗ is infinitely divisible by a prime q, then q appears in the sequence p⃗ and h has the form

h = waw^{−1}

for some a ∈ Gp⃗ ∪ Hp⃗.

Proof (a) Suppose h, g ∈ Γp⃗ and g^{2} = h. In general we will measure the length of a word in Γp⃗ by looking at the number of times it alternates between Gp⃗ and Hp⃗.

We write h as a reduced word in the form

h = w^{⌢}(a^{k})^{⌢}w^{−1}

where w is as long as possible and a is as short as possible. We write g as a reduced word

g = v^{⌢}b^{⌢}v^{−1}

where again v is as long as possible. Then it follows that

g^{2} = v^{⌢}(b^{2})^{⌢}v^{−1},

and hence v = w, b^{2} = a^{k}.

Now there are two cases.

In one case we have a ∉ Gp⃗ ∪ Hp⃗, when indeed if follows that b appears as an initial string in a^{k}, and hence, by minimization of a, that 2 divides k, and we have g = w^{⌢}(a^{k/2})^{⌢}w^{−1} as required.

The second case is a ∈ Gp⃗ ∪ Hp⃗. Assume a ∈ Gp⃗. Then since Gp⃗ is torsion free we have again that b = a^{k/2} is the unique element with b^{2} = a^{k}.

(b) If we have a prime q and h, g ∈ Γp⃗ with g^{q} = h then as above either (i) for some h̄, ḡ either both in Gp⃗ or both in Hp⃗ and some w ∈ Γp⃗ we have

h = w^{⌢}h̄^{⌢}w^{−1},
g = w^{⌢}ḡ^{⌢}w^{−1},

as reduced words and

ḡ + ḡ + ...(q times)... + ḡ = h̄

in the respective copy of ℚp⃗; or

(ii) for some b, w ∈ Γp⃗, b ∉ Gp⃗ ∪ Hp⃗ with w^{⌢}b^{⌢}w^{−1} reduced we have

h = w^{⌢}(b^{q})^{⌢}w^{−1},
g = w^{⌢}b^{⌢}w,

and as a reduced word

b^{q} = b^{⌢}b^{⌢}...(q times)...^{⌢}b.

The only possibility for infinite divisibility by q is the first case, when we indeed have to have q on the p⃗ sequence. □

Lemma 3.2 If e ∈ [S_{Γp⃗}], g ∈ Γp⃗, g · e = e (i.e. g ∈ stab{e}), g not the identity, then for some w, a ∈ Γp⃗, i ∈ ℤ, i ≠ 0,

e = w^{⌢}a^{⌢}a^{⌢}...
g = w^{⌢}(a^{i})^{⌢}w^{−1}.

Proof It is easily seen that e is eventually periodic, and so for some a, w

e = w^{⌢}a^{⌢}a^{⌢}...

Note then that

g · w^{⌢}a^{⌢}a^{⌢}... = w^{⌢}a^{⌢}a^{⌢}...

if and only if

w^{−1}gw · a^{⌢}a^{⌢}... = a^{⌢}a^{⌢}...

Thus we may as well assume that e is outright periodic and suppose that e has period n – where the period is the least n for which it is it is n-periodic. We may also assume without loss that there are g_{i}’s in Gp⃗ and h_{i}’s in Hp⃗ with

a = g_{1}^{⌢}h_{2}^{⌢}g_{3}^{⌢}...^{⌢}h_{n}.

We suppose g · e = e with

g = k_{1}^{⌢}k_{2}^{⌢}...^{⌢}k_{ℓ},

where, as usual, k_{i} ∈ Hp⃗ iff k_{i+1} ∈ Gp⃗ iff k_{i} ∉ Gp⃗.

Then by considering the fact that g · e = e still has period n either we obtain k_{ℓ} ∈ Hp⃗ and k_{ℓ} = h_{n}, and then k_{ℓ−1} = g_{n−1}, k_{ℓ−2} = h_{n−2}, and so on, until in the final end we have g = a^{i} some i > 0, or we obtain k_{ℓ} ∈ Gp⃗ and k_{ℓ} = g_{1}^{−1}, and then k_{ℓ−1} = h_{2}^{−1}, and it all unravels to the end with g = a^{−i} some i > 0. □

Corollary 3.3 If e_{1}, e_{2} ∈ [S_{Γp⃗}], then stab{e_{1}, e_{2}} is abelian.

Proof Whenever g ∈ stab{e_{1}, e_{2}} we have g^{2} is in stab{e_{1}} and thus we may appeal to 3.2 to write e_{1} as

e_{1} = w^{⌢}a^{⌢}a^{⌢}...

where lh(a) = period of e_{1} and g^{2} as

g^{2} = w^{⌢}(a^{i})^{⌢}w^{−1}.

Then by 3.1 we have that any such g will be in the cyclic subgroup generated w^{⌢}a^{⌢}w^{−1}. □

Lemma 3.4 There is a Borel Γp⃗ map

φ : [S_{Γp⃗}]^{3} → Γp⃗.

Proof Since Hp⃗, Gp⃗ are torsion free there are, by 1.4, respective Hp⃗- and Gp⃗-maps

θ_{H} : [Hp⃗]^{3} → Hp⃗,
θ_{G} : [Gp⃗]^{3} → Gp⃗.

Now for any {f_{1}, f_{2}, f_{3}} ∈ [S_{Γp⃗}]^{3} we follow C3.2 [1] and consider a split in cases.

Case(a) There is a reduced word v such that any two of the ends agree exactly up to v.

Thus we may write these as

f_{1} = v^{⌢}k_{1}^{⌢}k_{1}^{′⌢}...
f_{2} = v^{⌢}k_{2}^{⌢}k_{2}^{′⌢}...
f_{3} = v^{⌢}k_{3}^{⌢}k_{3}^{′⌢}...

where at each i we have k_{i} ∈ Hp⃗ iff k′_{i} ∈ Gp⃗ iff k_{i} ∉ Gp⃗. We then let

φ({f_{1}, f_{2}, f_{3}}) = vθ({k_{1}, k_{2}, k_{3}})

where θ = θ_{G} or θ = θ_{H} depending on which group the k_{i}’s exist in.

Case(b) Two of the ends agree with one another more than either does with the third.

So we may write {e_{1}, e_{2}, e_{3}} = {f_{1}, f_{2}, f_{3}} with

e_{1} = u^{⌢}w^{⌢}k_{1}...

e_{2} = u^{⌢}w^{⌢}k_{2}...
e_{3} = u^{⌢}ŵ^{⌢}...

as reduced words and w, ŵ having no initial segment in common.

We then let

φ({f_{1}, f_{2}, f_{3}}) = (u^{⌢}w)θ({k_{1}, k_{2}, 0})

where θ is again either θ_{G}, θ_{H} and 0 is the zero (abelian group identity) in the respective copy of ℚp⃗.

Note that case(b) can be converted into case(a) by acting on the triple by any reduced group element of the form

g = v^{⌢}k^{⌢}(w^{−1})^{⌢}u^{−1},

where k is in the same copy of ℚp⃗ as k_{1}, k_{2} but is not equal to either k_{1}^{−1}, k_{2}^{−1}. Thus to verify that φ is indeed a Γp⃗-map it suffices to check the invariance property for g in this form.

But here one has

φ({g · f_{1}, g · f_{2}, g · f_{3}}) = φ(v^{⌢}(kk_{1})^{⌢}..., v^{⌢}(kk_{2})^{⌢}..., v^{⌢}k^{⌢}(w^{−1})^{⌢}ŵ^{⌢}...)
= v · θ({kk_{1}, kk_{2}, k}) = v^{⌢}k · θ({k_{1}, k_{2}, 0})
= v^{⌢}k · ((w^{−1})^{⌢}u^{−1} · (u^{⌢}w · φ({k_{1}, k_{2}, 0}))) = v^{⌢}k^{⌢}(w^{−1})^{⌢}u^{−1} · θ({e_{1}, e_{2}, e_{3}}),

as required.

□

Definition Let H∞ ; p⃗ and G∞ ; p⃗ be the respective one point compactifications of Hp⃗ and Gp⃗.

So H∞ ; p⃗ consists of Hp⃗ along with a new point ∞_{H}, and the open sets are all sets which do not contain the point ∞_{H} along with all cofinite sets which do contain ∞_{H}. Similarly G∞ ; p⃗ contains the new point ∞_{G}.

We let Hp⃗ and Gp⃗ act on their one point compactifications so that the action extends the usual action on themselves by left translation and the new point at infinity becomes a fixed point – so

h · a = h + a

for h, a ∈ Hp⃗, whilst

h · ∞_{H} = ∞_{H}

for h ∈ Hp⃗, and similarly on the Gp⃗ side.

It is easily checked that these actions are continuous.

Definition We let Γp⃗^{∗} consist of all finite sequences

a_{0}^{⌢}a_{1}^{⌢}...a_{n}

where the a_{i}’s, i < n, alternate between Hp⃗ and Gp⃗, as in the definition of [S_{Γp⃗}], but the final point a_{n} is either ∞_{H} or ∞_{G}, depending on whether a_{n−1} is in Gp⃗ or Hp⃗.

Note that there is a natural analog of “concatenate and reduce” which defines an action of Hp⃗, or Gp⃗, on Γp⃗^{∗}: Given h ∈ Hp⃗ and w = a_{0}^{⌢}a_{1}^{⌢}a_{2}...^{⌢}a_{n} we define h · w by cases:

(i) if a_{0} ∈ G∞ ; p⃗, then

h · w = (h^{⌢}a_{0}^{⌢}a_{1}...a_{n});

(ii) if n = 0 and a_{n} = ∞_{H} then h · w just equals w again;

(iii) if n > 0, a_{0} ∈ Hp⃗, a_{0} ≠ −h, then
h · w = ((h + a_{0})^{⌢}a_{1}^{⌢}...a_{n});
(iv) if n > 0 and a_{0} = −h, then
h · w = (a_{1}^{⌢}a_{2}^{⌢}...a_{n}).

The action by Gp⃗ is the obvious analog, and we then obtain an action by the free product Γp⃗ = Gp⃗ ∗ Hp⃗.

Lemma 3.5 The stabilizers of points in the action of Γp⃗ on Γp⃗^{∗} are abelian.

Proof Consider some point w = a_{0}^{⌢}a_{1}^{⌢}...a_{n} ∈ Γp⃗^{∗}. We let u = a_{0}^{⌢}a_{1}...a_{n−1}, then the stabilizer of w in the action of Γp⃗ on Γp⃗^{∗} equals uHp⃗u^{−1} or uGp⃗u^{−1} depending on whether a_{n} equals ∞_{H} or ∞_{G}. □

Definition We let [S_{Γp⃗}]^{∗∞} be the union of [S_{Γp⃗}] and Γ∗ ; p⃗; thus it consists in all finite sequences alternating between G∞ ; p⃗, H∞ ; p⃗ and only taking the respective ∞_{G} or ∞_{H} at the final point along with all infinite sequences that alternate between Gp⃗ and Hp⃗.

We endow it with a topology as follows. Given a finite sequence a⃗ = a_{0}^{⌢}a_{1}^{⌢}...^{⌢}a_{n} in Γp⃗ and V an open subset of either H∞ ; p⃗ or G∞ ; p⃗, depending on whether a_{n} is in Gp⃗ or Hp⃗ respectively, we let

O(V, a⃗)

be the set of all b⃗ ∈ [S_{Γp⃗}] ∪ Γp⃗^{∗} where

b_{i} = a_{i}

all i ≤ n, and

b_{n+1} ∈ V.

We then equip [S_{Γp⃗}]^{∗∞} with the topology generated by taking the collection of all such O(V, a⃗) as our basis.

Lemma 3.6 [S_{Γp⃗}]^{∗∞} is a compact metric space.

Proof It is an easy exercise to check sequential compactness of the space directly by hand, which then suffices by second countability of the space; alternatively we may observe that [S_{Γp⃗}]^{∗∞} is the continuous image of a closed subset of the compact space

(H∞ ; p⃗ ∪ G∞ ; p⃗)^{ℕ}.

This second argument also serves to show that the space is separable.

Since [S_{Γp⃗}]^{∗∞} is zero dimensional and compact, any two disjoint closed sets can be separated by a clopen sets. As the space is obviously Hausdorff, we may then use Urysohn’s criterion to determine that the space is metrizable. □

Lemma 3.7 Γp⃗ is near-hyperbolic (in the sense of [1]).

Proof Following C3.3 [1] we consider the action on the compact metric space [S_{Γp⃗}]^{∗∞}. The equivalence relation induced on [S_{Γp⃗}] is hyperfinite since it is included in the tail equivalence relation, and there are only two orbits in the action of Γp⃗ on Γp⃗^{∗}. Thus certainly the induced orbit equivalence relation on the whole space [S_{Γp⃗}]^{∗∞} is hyperfinite and hence amenable, as is the induced orbit equivalence relation on M_{≤2}([S_{Γp⃗}]^{∗∞}).

From 3.3 and 3.5 we have that the stabilizers of points in M_{≤2}([S_{Γp⃗}]^{∗∞}) are abelian, and so certainly amenable. From 3.4 and 1.5 we obtain a Borel Γp⃗-map

η : M_{≥3}([S_{Γp⃗}]^{∗∞}) → [Γp⃗]^{<∞}.

Since Γp⃗ is torsion free we have from 1.4 that that the stabilizers of the action on [Γp⃗]^{<∞} and hence

M_{≥3}([S_{Γp⃗}]^{∗∞}) are trivial. It also follows that the induced orbit equivalence relation E_{Γp⃗}^{M_{≥3}([S_{Γp⃗}]^{∗∞})} is smooth, since {η^{−1}[F] : F ∈ [Γp⃗]^{<∞}} partitions the space into countably many Borel pieces, each of which meets a given orbit at most once. □

Lemma 3.8 Suppose p⃗, q⃗ are sequences of primes, Γp⃗, Γq⃗ are constructed as above, Xp⃗, Xq⃗ are standard Borel spaces, and that Kp⃗ = Γp⃗ × Γp⃗, Kq⃗ = Γq⃗ × Γq⃗ act freely and in a mixing and measure preserving manner on Xp⃗, Xq⃗, with the action of any non-amenable subgroup being E_{0}-ergodic. Then E_{Kp⃗} ≤_{B} E_{Kq⃗} implies p⃗ is included in q⃗.

Proof We consider the induced

E_{Γp⃗×ℤ} ⊂ E_{Kp⃗}

obtained by finding a copy of ℤ inside the right hand copy of Kp⃗. The reduction of E_{Kp⃗} to E_{Kq⃗} induces a countable to one homomorphism f of E_{Γp⃗×ℤ} to E_{Kq⃗} = E_{Γq⃗×Γq⃗} and we obtain induced cocycles

α_{1} : (Γp⃗ × ℤ) × Xp⃗ → Γq⃗,
α_{2} : (Γp⃗ × ℤ) × Xp⃗ → Γq⃗,

by considering the two copies of Γq⃗ which underpin E_{Kq⃗}. Note that we cannot have both of these cocycles equivalent to cocycles into amenable subgroups of Γq⃗ since E_{Γp⃗×ℤ} is E_{0}-ergodic.

The action of Kp⃗ = Γp⃗×Γp⃗ is mixing, and therefore every infinite subgroup acts ergodically. In particular for ℤ in the subgroup Γp⃗ × ℤ we have that ℤ acts in an ergodic fashion.

[f(x)]_{Γq⃗×Γq⃗}

is not a.e. constant. Thus 2.2 [1] applied to Γ = Γp⃗ × ℤ, Δ = ℤ ◁ Γ yields that one of α_{1}, α_{2} is equivalent to a cocycle induced by a homomorphism π : Γp⃗ × ℤ → Γq⃗ × Γq⃗ with non-amenable image. Since the image is in particular non-trivial it follows that there is some g ≠ 0 in Gp⃗ ∪ Hq⃗ with π(g) ≠ 0.

However g is divisible by every power of every prime on the p⃗ sequence, and hence so is π(g). Thus the lemma follows by 3.1. □

In the following lemma

p_{1}, p_{2} : Γp⃗ × Γp⃗ → Γp⃗

are the respective projections to the first and second copies of Γp⃗.

Lemma 3.9 Suppose Kp⃗ = Γp⃗ × Γp⃗ acts freely and in a measure preserving and mixing manner on the standard Borel probability space (X, μ) and the action of each non-amenable subgroup is E_{0}-ergodic. Let

α : Kp⃗ × X → Γp⃗ × Γp⃗(= Kp⃗)

be a Borel cocycle such that its restriction to any non-amenable subgroup is not equivalent to a cocycle taking values in an amenable subgroup of Kp⃗.

Then there is an infinite cyclic subgroup Δ of Kp⃗ such that the restricted cocycle

α : Δ × X → Γp⃗ × Γp⃗

is equivalent to a cocycle

α̂ : Δ × X → Γp⃗ × Γp⃗

such that (i) p_{1} ∘ α̂ : Δ × X → Γp⃗ given by a non-trivial homomorphism of Δ into Γp⃗ a.e; (ii) p_{2} ∘ α̂ : Δ × X → Γp⃗ is trivial a.e. (that is to say, p_{2} ∘ α̂(δ, x) is the identity μ a.e. x).

Proof We find some copy 𝔽_{2} = ⟨a, b⟩ of the free group inside Γp⃗ and consider the induced cocycles α_{1,j} = p_{j} ∘ α|_{(𝔽_{2}×⟨a⟩)×X} : (𝔽_{2} × ⟨a⟩) × X → Γp⃗,

α_{2,j} = p_{j} ∘ α|_{(⟨a⟩×𝔽_{2})×X} : (⟨a⟩ × 𝔽_{2}) × X → Γp⃗,

into the j^{th} copy of Γp⃗, for j = 1, 2. We let

β_{1,j} = α_{1,j}|_{({1}×⟨a⟩)×X} : ({1} × ⟨a⟩) × X → Γp⃗,

β_{2,j} = p_{j} ∘ α_{2,j}|_{(⟨a⟩×{1})×X} : (⟨a⟩ × {1}) × X → Γp⃗,

be the respective restrictions. Claim(1): For either j = 1 or j = 2 we have that the cocycle

β_{1,j} : ({1} × ⟨a⟩) × X → Γp⃗

is equivalent to a trivial cocycle (a.e.). Proof of claim: Or else 2.2 [1] gives that the induced cocycles

α_{1,j}|_{(𝔽_{2}×{1})×X} : (𝔽_{2} × {1}) × X → Γp⃗,

(j = 1, 2), are both equivalent a.e. to cocycles into amenable subgroups of Γp⃗ and hence the restricted cocycle

α|_{(𝔽_{2}×{1})×X} : (𝔽_{2} × {1}) × X → Kp⃗,

is equivalent to a cocycle a.e. into an amenable group, in contradiction to assumptions of lemma. (□Claim) Claim(2): For either ℓ = 1 or ℓ = 2

α_{2,ℓ} : (⟨a⟩ × 𝔽_{2}) × X → Γp⃗

is equivalent to a homomorphism with non-amenable image in Γp⃗. Proof of claim: Again by 2.2 [1]. (□Claim) Claim(3): Given j as in claim(1) and ℓ as in claim(2), ℓ ≠ j. Proof of Claim: Suppose instead ℓ = j. Then β_{1,ℓ} is equivalent to the trivial cocycle, which always returns the identity element of the group Γp⃗. On the other hand, the assumption on j = ℓ gives that

β_{1,ℓ} = α_{2,ℓ}|_{({1}×⟨a⟩)×X}

is equivalent to a cocycle by a non-trivial homomorphism, which will then necessarily be injective since Γp⃗ is torsion free.

Thus we have established that β_{1,ℓ} is simultaneously equivalent to the trivial cocycle and a cocycle given by an injective homomorphism, and this directly contradicts 3.5 [1]. (□Claim)

Thus we get an induced cocycle

α|_{({1}×⟨a⟩)×X} = (β_{1,1}, β_{1,2}) : ({1} × ⟨a⟩) × X → Γp⃗ × Γp⃗

as required. □ We these lemmas in place we can complete the proof of the messy fact.

Lemma 3.10 For any infinite string of primes p⃗ there is a corresponding standard Borel probability space space

(Xp⃗, μp⃗)

on which Kp⃗ acts in a mixing, measure preserving manner, with every non-amenable subgroup having an E_{0}-ergodic action and every infinite cyclic subgroup acting in a uniquely ergodic manner. This can be done so that

⋃_{p⃗}{p⃗} × Xp⃗

is a standard Borel space, the projection function

(p⃗, x) ↦ p⃗

is Borel, and the assignment of measures

p⃗ ↦ δp⃗ × μp⃗

is Borel as a function from infinite sequences of primes to M(⋃p⃗{p⃗} × Xp⃗).

Proof From A6.1, A4.1, and 3.7 of [1], but we do need to see that this assignment of Xp⃗ to p⃗ is effective. For each sequence of primes p⃗ let Xp⃗ ⊂ 2^{Kp⃗} be the collection of all

f : Kp⃗ → {0, 1}

such that for all infinite cyclic groups ⟨σ⟩ ⊂ Kp⃗ and clopen U ⊂ 2^{Kp⃗} we have

lim_{n→∞} \frac{1}{2n + 1}|\{i : −n ≤ i ≤ n, σ^{i} · f ∈ U\}| = μp⃗(U),

where μp⃗ is the usual Bernoulli “coin flipping” measure on 2^{Kp⃗}. Thus by the ergodic theorem (Xp⃗, μp⃗) is uniquely ergodic on all infinite subgroups. By A4.1 [1] every non-amenable subgroup gives rise to an E_{0}-ergodic equivalence relation. By A6.1 [1] the action is mixing.

The various conditions about the assignment of the space and the measure being suitably Borel in p⃗, as stated above in the lemma, are routinely verified since Xp⃗ is a Borel subset of 2^{Kp⃗} uniformly in p⃗. □

For x ∈ 2^{ℕ} we use almost disjoint coding to assign continuously an infinite sequence p⃗(x) of distinct primes such that for x ≠ y we have p⃗(x) and p⃗(y) eventually disjoint. We let

(X_{x}, μ_{x}, E_{x}) = (X_{p⃗(x)}, μ_{p⃗(x)}, E_{K_{p⃗(x)}});

we need to verify conditions (i)-(v) of the messy fact.

(i), (iv), and (v) are all immediate from the construction. (ii) follows from 3.8. For (iii), we suppose f : X_{x} → X_{x} is a homomorphism of E_{x} to E_{x}. Then 3.9 gives some infinite cyclic ⟨σ⟩, ⟨τ⟩ ⊂ K_{p⃗(x)} and measurable

θ : X_{x} → K_{p⃗(x)}

with

θ(σ · x) · f(σ · x) = τ · θ(x) · f(x).

Thus if we define

f̂: X_{x} → X_{x},
x ↦ θ(x) · f(x)

then this gives a measurable map which conjugates the action of σ with that of τ, and hence, by our unique ergodicity assumption, we have

μ_{x}(im(f̂)) > 0,
∴ μ_{x}(im(f)) > 0.

4 Is the notion of near hyperbolic interesting?

The definition of near hyperbolic was formulated in [1] as a purely ad hoc measure to help organize the proofs, and the example discussed there were in fact hyperbolic. In the present paper the examples given are not hyperbolic, and yet they would seem to natural instances of near hyperbolic groups.

Given that the methods of [1] work exactly for the class of groups which are near hyperbolic, it might be interesting to determine which groups lie inside that class, just as it might be interesting to determine which groups allow the kinds of rigidity theorems found in either [1] or [4]. In particular, we do not know what relationship exists between the class of near hyperbolic groups and the classes C and C_{reg} of Monod and Shalom in [4]. They prove that every group embeds into one in C_{reg} However while the notion of near hyperbolic goes down to subgroups, groups such as 𝔽_{2}×𝔽_{2} can not be near hyperbolic since, for instance, they do not satisfy the kinds of rigidity theorems shown in [1] for this class, and thus, by Monod and Shalom, not every C_{reg} group is near hyperbolic. This leaves open the reverse inclusion, just as they leave open whether C_{reg} is strictly included in C, and indeed the real task at this stage would seem to be to characterize the groups having varying levels of rigidity for their actions.

Here it is tempting to ask the following audacious question:

Question Let Γ_{1} and Γ_{2} be torsion free infinite groups, and let Γ_{1} × ℤ and Γ_{2} × ℤ act in a mixing and measure preserving manner on standard Borel probability spaces X_{1} and X_{2}. If E_{Γ_{1}} is orbit equivalent to E_{Γ_{1}}, must there exist a homomorphism from Γ_{1} to Γ_{2} with amenable kernel?

This, along with a great deal more, certainly holds when Γ_{1}, Γ_{2} are in the class of near hyperbolic by [1] or when they are in C by 2.23 [4]. However it is hard to imagine that the answer to the above question could possibly be affirmative, since the methods used in the known cases are so specific.

References

[1] G. Hjorth, A.S. Kechris, Rigidity theorems for actions of product groups and countable Borel equivalence relations, to appear in the Memoirs of the American Mathematical Society, available at

    http://www.math.caltech.edu/people/kechris.html.

[2] A.S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1994.

[3] S. Jackson, A.S. Kechris, A. Louveau, Countable Borel equivalence relations, Journal of Mathematical Logic, vol. 2(2002), pp. 1-80.

[4] N. Monod, Y. Shalom, Orbit equivalence and bounded cohomology, to appear in the Annals of Math-

    ematics, available at http://www.math.uchicago.edu/~monod/publications.html.

[5] S. Thomas, Superrigidity and countable Borel equivalence relations, Annals of Pure and Applied Logic, vol. 120 (2003), pp. 237-262.