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
(= 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
) 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
. 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|
Next fix i(δ) such that for all i i(δ) and |n| N,
3(2N + 1)
For such i, if
Ci = {x : ∃|n|
= Ti(x))},
then µ(Ci) δ/3.
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