9. INNER AMENABILITY 55

This is well-defined. Clearly

˜n(∆

T · x)EF2

Y

∆ · x and

˜n

T is 1-1 and measure

preserving on its domain, which has ν-measure equal to µ(∆ · An) → 1. So

we can extend

˜

T

n

to Sn ∈ [EF2

Y

].

We will check that {Sn} witnesses that [EF2

Y

] is not closed in N[EF2

Y

].

For this it is enough to verify:

(i) Sn → 1 weakly,

(ii) SnγSn

−1

→ γ uniformly, ∀γ ∈ F2,

(iii) δu(Sn, 1) → 0.

Now (i), (ii) easily follow from the fact that {Tn} has the same properties. To

prove (iii) we will use the following simple fact: If E is a smooth equivalence

relation and Un ∈ [E],Un → 1 weakly, then Un → 1 uniformly. This is

because w and u agree on [E], as [E] is weakly closed (or write X =

i

Ai,

where {Ai} is a pairwise disjoint family of partial Borel transversals and

note that µ({x ∈ Ai : Un(x) = x}) =

1

2

µ(Un(Ai)∆Ai)). Assume then that

(iii) fails, towards a contradiction. From the definition it follows then that

there is Un ∈ [E∆

X

] such that δu(Un,Tn) → 0 and so Un → 1 weakly (since

Tn → 1 weakly) and Un → 1 uniformly, a contradiction. ✷

Schmidt [Sc2] raised the question of whether there is a converse of 9.1

for ICC groups.

Problem 9.3. Assume that Γ is countable, ICC and inner amenable. Is

there a free, measure preserving, ergodic action of Γ for which [EΓ

X

] is not

closed in N[EΓ

X

]?

In connection with this question we note the following corollary of 8.1,

8.6.

Theorem 9.4 (Kechris). Suppose Γ is a countable group that admits a free,

measure preserving, ergodic action on (X, µ) with [EΓ

X

] not closed in N[EΓ

X

].

Then C(Γ) = 1. If moreover, Γ is strongly treeable (i.e., every equivalence

relation induced by a free, measure preserving action of Γ is treeable), then

Γ is amenable. Thus for strongly treeable groups Γ, Γ is amenable iff there

is a free, measure preserving, ergodic action of Γ on (X, µ), with [EΓ

X

] not

closed in N[EΓ

X

].

Note that, in view of 9.4, a positive answer to 9.3 implies that C(Γ) = 1

for any ICC inner amenable group, which is also an open problem.

Concerning 9.3, we also have the following partial answer.

Proposition 9.5. Let Γ be a countable ICC group which is weakly commu-

tative. Then there is a free, measure preserving, ergodic action of Γ with

[EΓ

X

] not closed in N[EΓ

X

].

Before we start the proof, let us notice certain general facts about con-

jugacy shift actions. Let Γ be a countable group, put

Γ∗

= Γ \ {1},

and consider its conjugacy shift action on

2Γ∗

: (γ · p)δ =

p(γ−1δγ).