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
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