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

2Γ

is not orbit equivalent to the conjugacy shift action on

2Γ∗

.

Proof. Since the shift action of Γ on

2Γ

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

2Γ

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