9. INNER AMENABILITY 57
Theorem 3.4 of Jones-Schmidt [JS] this is characterized by the existence
of a sequence {Tn} [EΓ
X
] for which we have Tn 1 in N[EΓ
X
] but for
some non-trivial asymptotically invariant sequence {An} MALGµ we have
µ(Tn(An)∆An) 0. (Here {An} is a non-trivial asymptotically invariant
sequence if µ(γ · An∆An) 0, ∀γ Γ, but µ(An)(1 µ(An)) 0.) Note
that for such {Tn},Tn 1 uniformly, so {Tn} is a witness to the fact that
[EΓ
X
] is not closed in N[EΓ
X
].
Proposition 9.8. Suppose there exist {γn}, {δn} Γ that witness weak
commutativity of Γ but γnδn = δnγn, ∀n. (For example, let Γ =
n
Γn, Γn
non-abelian.) Then the conjugacy shift action of Γ on
2Γ∗
is stable.
Proof. As in the proof of 9.5, γn 1 in N[EΓ]. Put An = {p
2Γ∗
:
p(δn) = 1}. Then γn · An = {p
2Γ∗
: p(γnδnγn
−1)
= 1}, and γnδnγn
−1
= δn,
so
µ(γn · An An) = µ(γn · An)µ(An) =
1
4
.
Thus µ(γn · An∆An) =
1
2
. Finally notice that µ(γ · An∆An) 0, ∀γ Γ, as
γ · An = An, if n is large enough.
We now have the following application, where two actions are orbit equiv-
alent if the induced equivalence relations are isomorphic (see Section 10,
(A)).
Proposition 9.9. Let {Γn} be a sequence of countable non-trivial groups at
least one of which is not amenable. Then the shift action of Γ =
n
Γn on

is not orbit equivalent to the conjugacy shift action on
2Γ∗
.
Proof. Since the shift action of Γ on

is E0-ergodic, it is enough to
show that the conjugacy shift action is not.
Case 1. Eventually the Γn are abelian. Then Γ = × Z, Z infinite
abelian. Consider

p f(p)
2Z∗
,
where f(p)(z) = p(1,z). Then if p, p are shift conjugate, clearly f(p) =
f(p ). Thus the conjugacy shift action of Γ on
2Γ∗
is not ergodic, as f
cannot be constant on a conull set.
Case 2. Infinitely many Γn are not abelian, say for n = n1 n2 . . . .
Choose ani , bni Γni that do not commute. Let γi = (1,...,ani , 1,... ),δi =
(1,...,bni , 1,... ). These satisfy the hypothesis of 9.8, so the conjugacy shift
action of Γ on
2Γ∗
is stable, thus not E0-ergodic.
Addendum. Recently Tsankov showed that for any inner amenable
group Γ the conjugacy shift action of Γ on
2Γ∗
is not E0-ergodic and thus
9.9 is true for any non-amenable, inner amenable group.
Kechris and Tsankov [KT] then studied more generally the action of a
countable group Γ on a countable set I and its relationship to the correspond-
ing shift action of Γ on
2I.
They proved that the following are equivalent: i)
The action of Γ on I is amenable (i.e., admits a finitely additive probability
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