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 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
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