28 I. MEASURE PRESERVING AUTOMORPHISMS

3.13, it is isomorphic to E, and thus, using πn, there is a Borel measure

preserving action an of Γ on An such that γn acts non-trivially and the

equivalence relation induced by an is contained in En. Taking the union

of these actions on the An’s gives an action on Γ on X such that every

γ = 1 acts non-trivially and the equivalence relation induced by this action

is included in E. This clearly gives an embedding of Γ into [E].

Since every amenable group can be clearly embedded into [E], by the

result of Ornstein-Weiss [OW] and 3.13, it is clear that every residually

amenable Γ embeds into [E]. Finally, the last assertion of the proposition is

clear by taking N to be the intersection of the kernels of all homomorphisms

from Γ into [E]. ✷

We also have the following partial converse. It is a special case of a

result of Robertson [Ro], concerning groups embeddable into the unitary

group of the hyperfinite II1 factor but we give below a direct ergodic theory

argument.

Below we recall that a Borel (not necessarily countable) equivalence

relation R on a standard Borel space Y is smooth if there is Borel map

f : Y → Z, Z a standard Borel space, such that xRy ⇔ f(x) = f(y). If

R is countable, this is equivalent to the existence of a Borel selector for R.

Finally, if E is countable, measure preserving on (X, µ), then E is called

smooth if its restriction to a co-null E-invariant Borel set is smooth in the

previous sense. It is then easy to see that E is smooth iff E is finite.

Proposition 4.14. Let E be an ergodic, hyperfinite equivalence relation. If

Γ is a countable group, Γ ≤ [E] and Γ has property (T), then Γ is residually

finite.

Proof. Let F ⊆ E be the equivalence relation induced by Γ.

Claim. F is smooth.

Granting this, we can complete the proof as follows. Since F is smooth

and measure preserving it must have finite classes a.e. Decompose then

X into countably many Γ-invariant Borel sets A1,A2,... such that the Γ-

orbits in An have cardinality n. Fixing a Borel linear ordering on X, we

can identify each Γ-orbit in An with {1, 2,...,n} and thus each Γ-orbit in

An gives rise to a homomorphism of Γ into Sn. Let γ ∈ Γ be different

from 1. Then for some n, γ acts non-trivially on An, so the homomorphism

corresponding to some Γ-orbit in An sends γ to something different from 1

in Sn. So Γ is residually finite.

Proof of the claim. We will use the following result of Schmidt-

Zimmer (see Zimmer [Zi1], 9.1.1): If a property (T) group Γ acts in an

ergodic, measure preserving way on a standard measure space (Y, ν) and

α : Γ × Y → Z is a Borel cocycle of this action, then α is a coboundary, i.e.,

α(γ, x) = f(γ · x) − f(x), for some Borel f : X → Z.

Fix a free Borel action (n, x) → n · x of Z on (X, µ) which generates

E. To show that F is smooth, it is enough to show that if (Y, ν) is one of