20 I. MEASURE PRESERVING AUTOMORPHISMS

4. The Reconstruction Theorem

(A) We will prove here another beautiful result of Dye that asserts that

any ergodic E is completely determined by [E] as an abstract group. Given

two countable Borel measure preserving equivalence relations E, F we say

that E, F are isomorphic, in symbols

E

∼

=

F,

if there is T ∈ Aut(X, µ) such that except on a null set

xEy ⇔ T(x)FT(y),

or, equivalently, T[E]T

−1

= [F ], i.e., [E], [F ] are conjugate in Aut(X, µ).

Theorem 4.1 (Dye). Let E, F be ergodic equivalence relations. Then the

following are equivalent:

(i) E

∼

= F .

(ii) [E], [F ] are conjugate in Aut(X, µ).

(iii) ([E],u), ([F ],u) are isomorphic as topological groups.

(iv) [E], [F ] are isomorphic as abstract groups.

Moreover, for any algebraic isomorphism f : [E] → [F ], there is unique

ϕ ∈ Aut(X, µ) with f(T) =

ϕTϕ−1,

∀T ∈ [E].

(B) In preparation for the proof, we prove some results that are also

useful for other purposes.

Lemma 4.2. Let E be ergodic. Then every element of [E] is a product of 5

commutators in [E].

Proof. We will need two sublemmas.

Sublemma 4.3. Let E be ergodic and let T ∈ [E] be periodic. Then T is a

commutator in [E] and the product of two involutions in [E].

Proof. We can assume that for some n ≥ 2 all T-orbits have exactly

n elements. Let A be a Borel selector for the T-orbits. Split A = A1 ∪ A2,

where µ(A1) = µ(A2) =

1

2

µ(A). Let Q ∈ [E] be an involution that sends

B1 =

n−1

k=0

T

k(A1)

to B2 =

n−1

k=0

T

k(A2)

and conjugates T|B1 to T

−1|B2.

Let T1 = T|B1 ∪ id|B2,T2 = id|B1 ∪ T|B2. Then QT2

−1Q

= T1 and T =

T1T2 = QT2

−1QT2

= [Q, T2

−1]

= Q(T2

−1QT2).

✷

Sublemma 4.4. Let E be ergodic. If T ∈ [E], then we can write T =

ST , where S ∈ [E],S = [U, V ] = U1V1 with U, V, U1,V1 ∈ [E],δu(T , 1) ≤

1

2

δu(T, 1),U1,V1 involutions and

supp(U) ∪ supp(V ) ∪ supp(U1) ∪ supp(V1) ∪ supp(T ) ⊆ supp(T).

Proof. We can assume that T is aperiodic (since on the periodic part

of T we can apply 4.3 and take T = 1). We can then find a Borel complete

section A for the T-orbits such that in the usual Z-order of each T-orbit the

distance between successive elements of A is at least 2, and A is unbounded

in each direction in every orbit (see, e.g., [KM], 6.7). Let T = TA be the