3. FULL GROUPS OF EQUIVALENCE RELATIONS 19

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

2n,

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

n

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) =

1

2n

card{s ∈

2n

: 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 L´ 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

[E]:

Theorem 3.17 (Giordano-Pestov [GP]). If E is an ergodic equivalence

relation, then E is hyperfinite iff ([E],u) is extremely amenable.