Theorem 3.13 (Dye). Let E, F be ergodic hyperfinite Borel equivalence
relations. Then there is T Aut(X, µ) such that for all x, y in a set of
measure 1,
xEy T(x)FT(y).
Since the hyperfinite E which preserve µ and are ergodic are exactly
the ones induced by ergodic T Aut(X, µ), we have for any two ergodic
T1,T2 Aut(X, µ), that [T1], [T2] are conjugate in Aut(X, µ). Also from 3.5
we have the next result.
Corollary 3.14. If E is ergodic, then for any ergodic T Aut(X, µ), [E]
contains a conjugate of T , i.e., [E] meets every conjugacy class in ERG.
Thus, up to conjugacy, there is only one full group of a hyperfinite
ergodic equivalence relation and it is the smallest (in terms of inclusion)
among all full groups of ergodic countable equivalence relations.
(F) We conclude with an application of 3.8 due to Giordano-Pestov.
Recall that a topological group G is extremely amenable if every continuous
action of G on a (Hausdorff) compact space has a fixed point.
Theorem 3.15 (Giordano-Pestov [GP]). Let E be an ergodic, hyperfinite
equivalence relation. Then ([E],u) is extremely amenable.
Corollary 3.16 (Giordano-Pestov [GP]). The group (Aut(X, µ),w) is
extremely amenable.
The corollary follows from 3.15, since, by 3.1, the identity map is a
continuous embedding of ([E],u) into (Aut(X, µ),w) with dense range.
Proof of 3.15. We take E = E0. Denote by Gn = S2n the finite
group of permutations of
which we identify with a subgroup of [E],
identifying π S2n with the element of [E] given by sˆx π(s)ˆx. Clearly
G1 G2 . . . and
Gn is dense in ([E0],u) by 3.8.
Now the metric δu restricted to Gn is clearly the Hamming metric on
Gn : δu(π, ρ) =
: π(s) = ρ(s)}, and therefore, by a result of
Maurey [Ma], the family (Gn,δu|Gn,µn), where µn = counting measure on
Gn, is a evy family and so, by Gromov-Milman [GM], ([E0],u) is a L´evy
group and thus extremely amenable (see, e.g., Pestov [Pe2]).
Giordano-Pestov [GP] also prove that, conversely, if for an ergodic E
the group ([E],u) is extremely amenable, then E is hyperfinite, so this gives
a nice characterization of the hyperfiniteness of E in terms of properties of
Theorem 3.17 (Giordano-Pestov [GP]). If E is an ergodic equivalence
relation, then E is hyperfinite iff ([E],u) is extremely amenable.
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