58 I. MEASURE PRESERVING AUTOMORPHISMS

measure), ii) the shift action of Γ on

2I

admits non-trivial almost invariant

sets (equivalently: admits a non-trivial asymptotically invariant sequence),

iii) the Koopman representation associated to the shift action, restricted to

the orthogonal of the constant functions, admits non-0 almost invariant vec-

tors (see Section 10, (C) for an explanation of these notions). In particular,

if these conditions hold, the shift action of Γ on

2I

is not orbit equivalent to

the shift action of Γ on

2Γ,

provided Γ is not amenable.

We finally note the following (almost) characterization of weakly com-

mutative groups.

Proposition 9.10. Let Γ be a countable group.

i) If Γ is a non-discrete (i.e., not closed) normal subgroup of a Polish

group, then Γ is weakly commutative.

ii) If Γ is weakly commutative with trivial center, then Γ is a non-discrete

normal subgroup of a Polish group.

Proof. i) Assume that Γ is a non-discrete normal subgroup of the Polish

group G. Consider the map

π : G → Aut(Γ),

π(g) = (γ →

gγg−1).

This is a Borel homomorphism, so it is continuous, where Aut(Γ) has the

pointwise convergence topology. Since Γ is not discrete in G, there is a

sequence {γn} ⊆ Γ \ {1} with γn → 1 (in G). So

∀δ∀∞n(γnδγn −1

= δ), i.e.,

∀δ∀∞n(γnδ

= δγn), where

∀∞

means “for all but finitely many”. Thus Γ is

weakly commutative.

ii) Consider the map

ρ : Γ → Aut(Γ) = G,

ρ(γ) = (δ →

γδγ−1).

Since Γ has trivial center, ρ is an isomorphism, so we can identify Γ with

ρ(Γ), which is a normal subgroup of G. Let {γn} ⊆ Γ\{1} witness the weak

commutativity of Γ. Then γn = 1,γn → 1 in G, so Γ is not discrete in G. ✷

Thus a centerless group is weakly commutative iff it is (up to algebraic

isomorphism) a non-discrete normal subgroup of a Polish group (or equiv-

alently a dense normal subgroup of a non-discrete Polish group). In the

case of groups with non-trivial center (which are automatically weakly com-

mutative) it is not clear if there is a simple characterization of when they

are non-discrete normal subgroups of Polish groups. Tsankov pointed out

that if Γ has non-trivial finite center and countable automorphism group

Aut(Γ), then Γ cannot be a non-discrete normal subgroup of a Polish group

G. Otherwise if π : G → Aut(Γ) is defined as in the proof of 9.10, i)

and γn are distinct in Γ with γn → 1, then π(γn) → 1 in Aut(Γ), which

is discrete, so π(γn) = 1 for all large enough n and thus the center of Γ

is infinite. For example, one can take Γ = ∆ × Z2, where ∆ is a simple