6. AUTOMORPHISM GROUPS OF EQUIVALENCE RELATIONS 43
graph(S) E, then δu(TST
−1,S)
UT
×T
(f) f
2.
2
On the other hand
T UT
×T
is clearly a Borel homomorphism from the Polish group N[E] into
the Polish group
U(L2(E,
M)), thus it is continuous, so if Tn 1 in N[E],
then UTn×Tn 1 in
U(L2(X,
M)). So T UT
×T
is a homeomorphism of
N[E] with a (necessarily closed) subgroup of
U(L2(X,
M)), and we have the
following fact.
Proposition 6.3. For aperiodic E, the map T UT
×T
from N[E] to
U(L2(E,
M)) is a (topological group) isomorphism of N[E] with a closed
subgroup of
U(L2(E,
M)).
Thus we can identify each T N[E] with the corresponding unitary
operator UT
×T
on
L2(E,
M).
(C) It follows from 4.1 that for every ergodic E, N[E] can be also iden-
tified with the automorphism group, Aut([E]), of the (abstract) group [E].
More precisely, every S N[E] gives rise to the automorphism T
STS−1
of [E] and every automorphism of the (abstract) group [E] is of this form for
a unique S N[E]. Next notice that [E] is the smallest non-trivial normal
subgroup of N[E]. Indeed, if K N[E] is another normal subgroup, then,
since [E] is simple (by 4.6), either [E] K or else [E] K = {1}. In the
latter case K C[E], the centralizer of [E] in Aut(X, µ), which by 4.11 is
trivial, so K = {1}.
Assume now E, F are ergodic and f : N[E] N[F ] is an (abstract)
group isomorphism. Then f sends [E] onto [F ] so, by 4.1, E

= F and there is
ϕ Aut(X, µ) such that f(T) = fϕ(T) =
ϕTϕ−1,
∀T [E]. Thus maps
[E] onto [F ] and therefore maps N[E] onto N[F ]. Consider f
−1
◦fϕ. This
is an automorphism of N[E] which is trivial on [E]. Since [E] is simple, non-
abelian, a theorem of Burnside (see Thomas [Tho], 1.2.8) asserts that every
automorphism of N[E] (which can be identified with the automorphism
group of [E]) is inner and thus there is ψ N[E] such that f
−1
fϕ(T) =
ψTψ−1,
∀T N[E], and as f
−1
fϕ(T) = T for T
[E],ψTψ−1
= T, ∀T
[E], i.e., ψ C[E], so ψ = 1. Thus f(T) = fϕ(T), ∀T N[E].
So we have the following Reconstruction Theorem for N[E].
Theorem 6.4. If E, F are ergodic equivalence relations, the following are
equivalent:
(i) E

= F ,
(ii) N[E],N[F ] are isomorphic as abstract groups.
Moreover, for any algebraic isomorphism f : N[E] N[F ] there is
unique ϕ Aut(X, µ) with f(T) =
ϕTϕ−1,
∀T N[E].
(D) Finally, we note that Danilenko [Da] has shown that (as a topolog-
ical group) N[E] is contractible, when E is ergodic, hyperfinite. We will see
some further properties of such N[E] in the next section.
Comments. For the definition of the topology on N[E], see Hamachi-
Osikawa [HO], Gefter [Ge], Danilenko [Da].
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