4. THE RECONSTRUCTION THEOREM 27

of Schreier-Ulam [SU], that the set of pairs (g, h) ∈

U(n)2

that generate a

dense subgroup of U(n) is dense Gδ in

U(n)2.

The same result for U(H)

follows, using the Baire Category Theorem, since for each u ∈

n

U(n) and

open nbhd N of u, the set of (g, h) ∈

U(H)2

that generate a subgroup

intersecting N is open dense.

It is perhaps worth pointing out here that although, as we have seen

above, ([E],u) is topologically finitely generated, when E is given by an

ergodic action of a finitely generated group, for any E and for any n ≥ 1

it is not the case that the set of n-tuples (T1,...,Tn) ∈

[E]n

that generate

a dense subgroup of ([E],u) is dense in

[E]n

(with the product uniform

topology). To see this, simply notice that if T1,...,Tn satisfy du(Ti, 1)

for i = 1,...,n, then there is a set A ⊆ X with µ(A) 1 − n such

that Ti|A = id, ∀i ≤ n, so T|A = id for each T ∈ T1,...,Tn , thus if

n 1, T1,...,Tn ⊆ {T ∈ [E] : du(T, 1) n} is not dense in ([E],u).

This simple argument also limits the kinds of Polish groups that can

be closed subgroups of ([E],u) for an equivalence relation E. Call a Polish

group G locally topologically finitely generated if there is n ≥ 1 such that for

any open nbhd V of 1 ∈ G, there are g1,...,gn ∈ G with g1,...,gn dense in

G. Examples of such groups include

Rn, Tn,U(H),

Aut(X, µ). Also factors

and finite products of locally topologically finitely generated groups have

the same property. Now notice that by the above argument any continuous

homomorphism of such a group into the full group ([E],u) must be trivial. In

particular, a non-trivial locally topologically finitely generated group cannot

be a closed subgroup of ([E],u) (or even continuously embed into ([E],u)).

(E) Recently Pestov (private communication) raised the following re-

lated question: Let E be a measure preserving, ergodic, hyperfinite equiv-

alence relation. What kind of countable groups embed (algebraically) into

[E]? In response to this we mention the following two facts. For the first, re-

call that a countable group Γ is residually finite (resp., residually amenable)

if for any γ ∈ Γ,γ = 1 there is an homomorphism π : Γ → ∆, where ∆ is

finite (resp., amenable) such that γ ∈ ker(π). The following simple result

was originally proved for residually finite groups. Ben Miller then noticed

that the argument really shows the following stronger fact.

Proposition 4.13. Let E be an ergodic, hyperfinite equivalence relation.

Given a countable group Γ, if for every γ ∈ Γ\{1} there is a homomorphism

π : Γ → [E] such that γ ∈ ker(π), then Γ embeds into [E]. In particular

every residually amenable Γ embeds into [E] and for every countable group

Γ there is unique normal subgroup N such that Γ/N embeds into [E] and

every homomorphism from N into [E] is trivial.

Proof. Suppose (X, µ) be the space on which E lives. Let Γ \ {1} =

{γ1,γ2,... }. Fix then a sequence of homomorphisms πn : Γ → [E] such

that γn ∈ ker(πn). Next split X into countably many Borel sets A1,A2,...,

each meeting every E-class. Let En = E|An. Considering the space An with

the normalized restriction of µ to An, En is hyperfinite and ergodic, so, by