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 EΓ

X

= E.

We denote by CΓ 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 ∈ CΓ iff T is an isomorphism of

the action. In particular, CΓ ≤ N[E]. By Proposition 6.2, assuming E

is aperiodic, τN[E]|CΓ = w|CΓ. Thus, since CΓ is obviously a closed sub-

group of (Aut(X, µ),w), and therefore Polish, CΓ is a closed subgroup of

(N[E],τN[E]).

Now CΓ ≤ 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 ∈ CΓ 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 CΓ [E] (for the conjugation action), which we take

here to be the space CΓ × [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 : CΓ [E] → N[E]

be defined by

ϕ(T, S) = TS.

This is a continuous homomorphism of the Polish group CΓ E into the

group (N[E],τN[E]) whose range is the group

CΓ[E]

generated by CΓ and [E]. In particular, CΓ[E] is Polishable.

In the case CΓ ∩ [E] = {1} (and this is quite common, see 14.6 below),

then ϕ is a continuous injective homomorphism of CΓ [E] onto CΓ[E].

Thus if additionally CΓ[E] is closed, ϕ is a topological group isomorphism

of CΓ [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