48 I. MEASURE PRESERVING AUTOMORPHISMS

obtained by Gefter [Ge], Gefter-Golodets [GG], Furman [Fu1], Popa [Po4],

Ioana-Peterson-Popa [JPP] and Popa-Vaes [PV]. Among them are examples

where Out(E) is trivial, i.e., N[E] = [E].

8. Costs and the outer automorphism group

(A) We establish here a connection between the cost of an equivalence

relation and the structure of its outer automorphism group. Concerning the

theory of costs, see Gaboriau [Ga]. We follow in this section the terminology

and notation of Kechris-Miller [KM], Ch. 3. In particular, the cost of an

equivalence relation E on (X, µ) is denoted by Cµ(E).

Theorem 8.1 (Kechris). If the outer automorphism group of an ergodic

equivalence relation E is not Polish (i.e., if [E] is not closed in N[E]), then

Cµ(E) = 1.

Proof. To say that [E] is not closed in N[E] means that the identity

map from ([E],u) into N[E] is not a homeomorphism, i.e., the identity

map from ([E],τN[E]) into ([E],u) is not continuous, or, equivalently, that

there is a sequence {Tn} ⊆ [E] such that δu(TnT, TTn) → 0, ∀T ∈ [E], but

δu(Tn, 1) → 0.

Call a sequence {Tn} ⊆ [E] good if ∃ 0∀n(δu(Tn, 1) ≥ ) & ∀T ∈

[E](δu(TnT, TTn) → 0). Thus [E] is not closed in N[E] iff there is a good

sequence. Note also that a subsequence of a good sequence is good.

Lemma 8.2. If {Tn} is good, then E

Tn

n

(= the equivalence relation induced

by {Tn}) is aperiodic.

We will assume this and complete the proof.

Lemma 8.3. If {Tn} is good, S ∈ [E] is aperiodic, and 0, then there

is a sequence i0 i1 i2 . . . such that Cµ(E

S,Tik

) 1 + for any

subsequence ik0 ik1 ik2 . . . .

Proof. We first show that for any δ 0 there is i(δ) such that for

any i ≥ i(δ) there is Ai with µ(Ai) δ and {S, Ti|Ai} is an L-graphing of

E

S,Ti

. Since S is aperiodic, we can find a complete Borel section A of ES

with µ(A) δ/3. Then we can find large N so that µ(B) δ/3, where

B = {x ∈ X : ∀|n| ≤

N(Sn(x)

∈ A)}.

Next fix i(δ) such that for all i ≥ i(δ) and |n| ≤ N,

δu(TiSn,SnTi)

=

δu(S−nTiSn,Ti)

δ

3(2N + 1)

.

For such i, if

Ci = {x : ∃|n| ≤

N(S−nTiSn(x)

= Ti(x))},

then µ(Ci) δ/3.