Borel equivalence relations which are highly unfree^{∗}

Greg Hjorth

Abstract

There is an ergodic, measure preserving, countable Borel equivalence relation E on a standard Borel probability space (X, μ) such that E|_{C} is not essentially free on any conull C ⊂ X.

1 introduction

The results in this note are a direct response to developments from [6] and [7]. Following these papers we now understand much better the extent to which Borel equivalence relations may be reduced to Borel equivalence relations arising from a free group action.

Definition If E, F are Borel equivalence relations on standard Borel spaces X, Y (that is to say, they are Borel as subsets of X × X and Y × Y), then we write

E ≤_{B} F,

E Borel reducible to F, if there is a Borel φ : X → Y with

x_{1}Ex_{2} ⇔ φ(x_{1})F φ(x_{2}).

A Borel equivalence relation is countable if every equivalence is countable. (For a discussion of the general theory of countable Borel equivalence relations the reader should consult [5].) An equivalence relation E on a standard Borel probability space (X, μ) is measure preserving if every Borel partial bijection f : A → B included in the graph of E is measure preserving. The equivalence relation is ergodic if every E-invariant Borel set is either null or conull.

Theorem 1.1 (Feldman-Moore; [2]) Let E be a countable Borel equivalence relation on a standard Borel space X. Then there is a countable group acting by Borel automorphisms whose orbit equivalence relation equals E.

The authors of [2] go on to ask whether every such equivalence relation can be obtain be a free action of a countable group. Adams in [1] presented a negative solution to their question, but the proof worked, in some sense, by placing two mutually orthogonal measures on the space in question with respect to which the equivalence relation displayed very different measure theoretic properties.

One way to describe the example as is follows. Take a non-amenable group such as 𝔽_{2}, the free group on two generators, and let it act freely and by measure preserving transformations on a standard Borel probability space (X_{0}μ_{0}) with induced equivalence relation F_{1}. Then let ℤ act freely and by measure preserving transformations on a standard Borel probability space (X_{1}, ν_{1}) with induced equivalence relation F_{2}. Let X be the disjoint union of X_{0} and X_{1} and let E be the disjoint union of F_{0} and F_{1}. The action of 𝔽_{2} on X_{0} is non-amenable in the sense of §4.3 [8] simply because 𝔽_{2} is non-amenable, and hence the orbit equivalence relation can only be induced by a measurable action of a non-amenable group in light of 4.3.3 [8]; on the other hand, the action of ℤ is amenable, and any group acting freely to give rise to the same orbit equivalence relation must itself be amenable, again by 4.3.3 [8].

∗Key words and phrases: Ergodic theory, Borel equivalence relations, free group actions, Borel reducibility. AMS Subject Classification: 03E15, 28D15, 37A15.

Adams was to led to recast a measure theoretic version of the question from [2] – considering only measure preserving, ergodic equivalence relations and then only asking for a free action after discarding a null set. A negative answer was obtained in [3]:

Theorem 1.2 (Furman; [3]) There is an ergodic, measure preserving, countable Borel equivalence relation E on a standard Borel probability space (X, μ) such that on any conull C ⊂ X we have E|_{C} not induced by the free, Borel action of a countable group.

Definition (Thomas) A countable Borel equivalence relation E on a standard Borel space is essentially free if it is Borel reducible to an equivalence relation arising from the free Borel action of a countable group.

Using the results of [6]:

Theorem 1.3 (Thomas; [7]) Not every countable Borel equivalence relation is essentially free.

Moreover [7] also shows there to be no maximum element in the essentially free Borel equivalence relations under ≤_{B}.

However Thomas was prompted to phrase Adams’ question in to the new context, asking whether every countable Borel, ergodic, measure preserving, equivalence relation is on some conull subset Borel reducible to the free Borel action of a countable group.

Theorem 1.4 There is an ergodic, measure preserving, countable Borel equivalence relation E on a standard Borel probability space (X, μ) such that E|_{C} is not essentially free on any conull C ⊂ X.

2 Acknowledgements

I am exceptionally grateful to Simon Thomas, who read through several earlier drafts of this note, pointed out a number off errors, and vastly improved the exposition.

3 Proofs

Definition Given S ⊂ ℤ define G_{S} to be the group with generators

a,
b_{i}, c_{i},

i ∈ ℤ, subject to the relations

ab_{i}a^{−1} = b_{i+1},
ac_{i}a^{−1} = c_{i+1},

and

b_{i}c_{i+j} = c_{i+j}b_{i}

for j ∈ S.

Lemma 3.1 For i, j ∈ ℤ, if b_{i}c_{i+j} = c_{i+j}b_{i} in G_{S} then j ∈ S.

Proof First let K_{S} be the group generated by just the b_{i}’s and c_{i}’s subject to the relation b_{i}c_{i+j} = c_{i+j}b_{i} for j ∈ S. Not that for any i, i + j with j ∉ S we can define

π : K_{S} → 𝔽_{2} = ⟨σ, τ⟩

b_{i} ↦ σ,

c_{i+j} ↦ τ,

and all other generators get sent to the identity e. It is easily checked that this is a homomorphism, and thus b_{i} and c_{i+j} do not commute in K_{S}.

Now we can define a homomorphism

φ : ℤ → Aut(K_{S})
n ↦ φ_{n}

where

φ_{n} : h ↦ a^{n}ha^{−n}.

This is a homomorphism and we immediately have G_{S} isomorphic to the semi-direct product

ℤ ⋊ K_{S},

and hence b_{i}c_{i+j}b_{i}^{−1}c_{i+j}^{−1} is not equal to the identity in G_{S}. □

Lemma 3.2 Each G_{S} is finitely generated.

Proof a, b_{0}, c_{0} generates. □

Lemma 3.3 The canonical map a ↦ a, b_{0} ↦ b_{0}, c_{0} ↦ c_{0} generates an isomorphism from G_{S} to G_{T} if and only if S = T.

Lemma 3.4 Each G_{S} has no non-trivial finite normal subgroups.

Proof Clearly a is always torsion free, so no finite subgroup can contain this element. However, the other elements in the group all have infinite image under the action of conjugation by elements of the cyclic subgroup ⟨a⟩. □

Proof From 3.1, 3.2. □

Lemma 3.5 If H is a finitely generated group, then there are only countably many S’s for which there is a homomorphism

π : G_{S} → H

with finite kernel.

Proof By 3.4 we need only show that there only countably many S’s which admit a monomorphism into H. There are only countably many choices for π(a), π(b_{0}), π(c_{0}), and each of these determine π by 3.2. If

π_{S} : G_{S} → H,
π_{T} : G_{T} → H,

have the same values for π_{S}(a), π_{S}(b_{0}), π_{S}(c_{0}) and π_{T} (a), π_{T} (b_{0}), π_{T} (c_{0}) then 3.1 yields S = T. □

Definition For S ⊂ ℤ and n ∈ ℤ we let

n + S = {ℓ + n : ℓ ∈ S}.

S is periodic if for some n we have n + S = S. For S ⊂ ℤ let H_{S} be the group

SL_{3}(ℤ) × G_{S}.

Let X be the set of all pairs (S, f) where f ∈ 2^{H_{S}} and S is not periodic.

Given n ∈ ℤ, S, T ⊂ ℤ with n + S = T, we let

π_{S,T} : H_{S} ≅ H_{T}

induced by the canonical isomorphism

G_{S} ≅ G_{T} ,

a ↦ a,

b_{0} ↦ b_{0},
c_{0} ↦ c_{n}.

We will be only considering non-periodic subsets of the integers, and so the choice of the n will likewise be canonical.

We then let

σ_{S,T} : 2^{H_{S}} ≅ 2^{H_{T}}

be the canonical isomorphism

σ_{S,T} (f)(γ) = f(π_{S,T}^{−1} (γ)).

We then define E on X by

(S, f)E(T, g)

if for some (necessarily unique) n we have n + S = T and

σ_{S,T} (f)E_{H_{T}} g

in the space 2^{H_{T}}.

We let μ be the canonical measure on 2^{ℤ}, which we identify with P(ℤ). Note that the induced action of ℤ on 2^{ℤ} given by

(n, S) ↦ n + S

is μ-ergodic.

For each S ⊂ ℤ we let m_{S} be the canonical product measure on 2^{H_{S}}. Then for A ⊂ X Borel we let

ν(A) = ∫_{2^{ℤ}}(∫_{2^{H_{S}}} A_{S}dm_{S})dμ,

where A_{S} = {f ∈ 2^{H_{S}} : (S, f) ∈ A}. In other words, we define a measure on X by integrating the m_{S}’s against μ.

Lemma 3.6 E is a ν-ergodic, ν-invariant, countable Borel equivalence relation.

Lemma 3.7 Given any countable H there is a conull subset of 2^{ℤ} on which the resulting H_{S} does not admit a homomorphism to H with finite kernel.

Proof By 3.5. □

Corollary 3.8 For almost every S there is a conull subset of T ∈ 2^{ℤ} for which the resulting H_{T} does not admit a homomorphism from H_{S} to H_{T} with finite kernel.

Proof By 3.7 and Fubini’s theorem. □

We will use the following result, which was obtained in [7] from the Popa superrigidity results of [6].

Theorem 3.9 (Thomas; [7]) Let G be a countable group and let H = SL_{3}(ℤ) × G act by shift on 2^{H}. Let B ⊂ 2^{H} be a conull set. If K is some other group acting freely by Borel automorphisms on a standard Borel space Y with

E_{H}^{B} ≤_{B} E_{K}^{Y}

then there is a homomorphism from H to K with finite kernel.

Lemma 3.10 If B ⊂ X is conull, then E|_{B} is not essentially free.

Proof Suppose

θ : B → Y

reduces E|_{B} to E_{H}^{Y} where H is a countable group annd Y is a free Borel H-space. Then, by Fubini, we obtain uncountably many S’s for which B_{S}, by which I mean

{f ∈ 2^{H_{S}} : (S, f) ∈ B},

is conull in 2^{H_{S}}. Thus by 3.9 there are uncountably many S with a finite to one homomorphism

ϕ_{S} : G_{S} → H,

contradicting 3.5. □

This answers question 3.23 from [7].

Lemma 3.11 If X is written as the disjoint union smooth union of E-invariant sets,

X = ˙ ⋃_{z∈ℝ}X_{z},

then some X_{z} is not essentially free.

Proof Instead suppose

f : X → ℝ

is a Borel E-invariant function, with

X_{z} = f^{−1}[{z}.

Since E is ν-ergodic, some f^{−1}[{z}] is conull, and then the lemma follows from 3.10. □

Note in particular that our E is Borel reducible to E_{∞}. Thus we obtain from E not being reducible to a smooth union of disjoint free countable Borel equivalence relations that E_{∞} is likewise not reducible to a smooth union of disjoint essentially free countable Borel equivalence relations, thereby answering question 3.19 of [7].

There is perhaps something else interesting.

Lemma 3.12 For any conull B ⊂ X there exists a further conull C ⊂ B with E|_{B} not Borel reducible to E|_{C}.

Proof Choose S sufficiently random to ensure B_{S} = {f : (S, f) ∈ B} is conull in 2^{H_{S}} and that the conclusion of 3.8 holds for S. Choose C_{0} ⊂ 2^{ℤ} so that H_{S} does not admit a finite to one homomorphism to any H_{T} for T ∈ C_{0}.

Recall that at each T ∈ 2^{ℤ} there will be a conull subset of 2^{H_{T}} on which H_{T} acts freely. Let C be the

{(T, f) ∈ B : T ∈ C_{0}, f is in the free part of 2^{H_{T}} }.

Suppose

E|_{B} ≤_{B} E|_{C}

for a contradiction. In particular we can obtain some

θ : B_{S} → C

witnessing E_{H_{S}}^{B_{S}} ≤_{B} E|_{C}. Use θ to define

φ : B_{S} → 2^{ℤ}

by postcomposing with projection to the first coordinate – so that θ(f) will always have the form (φ(f), g) for some g. In particular we have φ is a homomorphism of equivalence relations, in the sense that if f_{1}E_{H_{S}} f_{2}, then φ(f_{1}) is equivalent to φ(f_{2}) under the shift action of ℤ on 2^{ℤ}.

As discussed in §A4 of [4], the orbit equivalence relation induced by H_{S} acting on B_{S} is E_{0}-ergodic, while the the orbit equivalence relation induced by ℤ on 2^{ℤ} is Borel reducible to E_{0}. Thus we may find a conull

D_{S} ⊂ B_{S}

and a single

T_{0} ∈ 2^{ℤ}

such that for all f ∈ D_{S}

φ(f)E_{ℤ}T_{0}.

Thus after possibly adjusting the Borel reduction θ we obtain

E_{H_{S}}^{D_{S}} ≤_{B} E_{H_{T_{0}}}^{C_{T_{0}}} ,

with a contradiction to 3.9. □

I did not know of any earlier example of an ergodic countable equivalence relation with this property, however this equivalence relation is itself not induced by the free action of a countable group. It seems then that there is an obvious further question:

Question Does there exist a free, Borel, measure preserving, ergodic action of a countable group H on a standard Borel probability space such that for any conull set B there exists a conull C ⊂ B with E_{H}|_{B} not Borel reducible to E_{H}|_{C}?

References

[1] S. Adams, An equivalence relation that is not freely generated. Proc. Amer. Math. Soc. vol. 102 (1988), 565–566

[2] J Feldman, C.C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, II Trans. Amer. Math. Soc. 234 (1977), 289–324, 325–359.

[3] A. Furman, Orbit equivalence rigidity, Ann. of Math. vol. 150 (1999).

[4] G. Hjorth, A.S. Kechris, Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations, Memoirs of the AMS, vol. 177, 2005.

[5] S. Jackson, A. S. Kechris, A. Louveau, Countable Borel equivalence relations, J. Math. Log., vol. 2 (2002), no. 1, 1–80.

[6] S. Popa, Cocycle and Orbit Equivalence Superrigidity for Bernoulli Actions of Kazhdan Groups, preprint.

[7] S. Thomas, Popa superrigidity and countable Borel equivalence relations, preprint.

[8] R.J. Zimmer, Ergodic Theory and Semisimple Groups, Birkh user, Basel, 1984.

greg.hjorth@gmail.com

http://www.math.ucla.edu/˜ greg/

Department of Mathematics and Statistics University of Melbourne Parkville 3010 Victoria Australia