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∗
: · p)δ =
p(γ−1δγ).
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