54 I. MEASURE PRESERVING AUTOMORPHISMS
so
α(n,
γ−1
·
x)γ−1
· x =
γ−1α(n,
x) · x
and, as the action is free,
α(n,
γ−1
·
x)γ−1
=
γ−1α(n,
x),
so
γ−1α(n,
x)γ = α(n,
γ−1
· x).
Thus
ϕ(γ · f) = lim
n→U
An
f(γ−1α(n,
x)γ)dµ(x) + lim
n→U
X\An
f(α(n,
γ−1
· x))dµ(x).
The second summand is equal to
lim
n→U
γ−1·(X\An)
f(α(n, x))dµ(x) = ϕ(f) lim
n→U
γ−1·An
f(α(n, x))dµ(x).
Thus
ϕ(γ · f) ϕ(f) = lim
n→U
An
f(γ−1α(n,
x)γ)dµ(x)−
lim
n→U
γ−1·An
f(α(n, x))dµ(x),
which equals 0, since µ(An) 0.
It follows from 9.1 that for every free, measure preserving action of a
countable group Γ, which is ICC and has property (T), [EΓ
X
] is closed in
N[EΓ
X]
(see Gefter-Golodets [GG]).
(C) The converse of 9.1 is literally false as stated, in view of the following
simple fact.
Proposition 9.2. If Γ = × F2, where is non-trivial finite, then for
every free, measure preserving, ergodic action on (X, µ), [EΓ
X
] is closed in
N[EΓ
X].
Proof. This follows from 8.1, as the cost C(Γ) is greater than 1 and Γ
has fixed price but one can also give a direct proof.
Let E =
X
. Let Y = X/∆ (we view here as a subgroup of Γ
and similarly for F2). Let π : X Y be the canonical projection and put
ν = π∗µ. Then F2 acts on Y by γ ·(∆·x) = ∆·(γ ·x). This is a free, measure
preserving, ergodic action of F2 on Y . If [EΓ
X
] is not closed in N[EΓ
X
], we
will show that [EF2
Y
] is not closed in N[EF2
Y
], contradicting 9.1.
Let Tn [EΓ
X
] be such that δu(Tn, 1) 0 but Tn 1 in N[EΓ
X
]. Thus
if An = {x : ∀δ ∆(Tn(δ · x) = δ · Tn(x))}, then µ(An) 1. Then for every
· x Y such that · x An = define
˜n(∆
T · x) = · Tn(y), for y · x An.
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