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 = EΓ

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.