7. THE OUTER AUTOMORPHISM GROUP 47
is inner. Also I do not know whether the following is true: If F is ergodic
and Out(F ) is algebraically isomorphic to Out(E), then F is hyperfinite.
Bezuglyi and Golodets [BG2] have generalized the Connes-Krieger the-
orem to show that if π1,π2 : G N[E] are two homomorphisms of a count-
able amenable group into N[E] and πi : G Out(E) is defined as πi = ϕ◦πi,
with ϕ : N[E] Out(E) the canonical surjection, then π1, π2 are conjugate
(i.e., ∃θ Out(E) with π1(g) =
θπ2(g)θ−1,
∀g G) iff ker(π1) = ker(π2).
Remark. Consider now a Borel measure preserving action of a count-
able group Γ on (X, µ) and the associated equivalence relation
X
= E.
We denote by the stabilizer of the action, i.e., the set of T Aut(X, µ)
such that (writing γ(x) = γ · x) TγT
−1
= γ, ∀γ Γ. This is equivalent
to T(γ · x) = γ · T(x), ∀γ Γ, i.e., T iff T is an isomorphism of
the action. In particular, N[E]. By Proposition 6.2, assuming E
is aperiodic, τN[E]|CΓ = w|CΓ. Thus, since is obviously a closed sub-
group of (Aut(X, µ),w), and therefore Polish, is a closed subgroup of
(N[E],τN[E]).
Now N[E] acts by conjugation on [E]. Since (CΓ,w) = (CΓ,τN[E])
is a closed subgroup of Iso([E],δu), when we identify T with S
[E] TST
−1
[E], and the evaluation action of Iso([E],δu) (with the
pointwise convergence topology) on ([E],u) is continuous, it follows that
the conjugation action of (CΓ,w) on ([E],u) is continuous. Consider then
the semidirect product [E] (for the conjugation action), which we take
here to be the space × [E] with the product topology and multiplication
defined by
(T1,S1)(T2,S2) = (T1T2, (T2
−1S1T2)S2)
(see Appendix I, (B)). This is again a Polish group. Let
ϕ : [E] N[E]
be defined by
ϕ(T, S) = TS.
This is a continuous homomorphism of the Polish group E into the
group (N[E],τN[E]) whose range is the group
CΓ[E]
generated by and [E]. In particular, CΓ[E] is Polishable.
In the case [E] = {1} (and this is quite common, see 14.6 below),
then ϕ is a continuous injective homomorphism of [E] onto CΓ[E].
Thus if additionally CΓ[E] is closed, ϕ is a topological group isomorphism
of [E] with (CΓ[E],τN[E]). Since ϕ({1} × [E]) = [E], we conclude that
[E] is closed in N[E]. Thus
(CΓ [E] = {1} and CΓ[E] is closed in N[E]) [E] is closed in N[E].
Comments. Very interesting explicit calculations of the outer automor-
phism groups of equivalence relations induced by certain actions have been
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