18 I. MEASURE PRESERVING AUTOMORPHISMS

(since S0,T0 are aperiodic). By 3.10, we can also assume that

A0,w1

S0,T0

(A0),...,wk0−1

S0,T0

(A0)

are pairwise disjoint and if the first symbol of wk0 is say S0

±1

(the other case

being similar), then S0

±1(wk0−1

S0,T0

(A0)) = wi

S0,T0

(A0), for some 0 ≤ i ≤ k0 − 2.

Finally, we can assume that µ(A0) /2.

If the first symbol of wk0 is S0 (the other case being similar), then by

3.11, applied to the set A = wi

S0,T0

(A0) and T = S0

−1,

we can find the

appropriate T , A ⊆ A and then, letting S0 = (T

)−1,A0

= (wi

S0,T0

)−1(A

)

we have an aperiodic S0 with δu(S0,S0) /2 and A0 ⊆ A0,µ(A0) 0 such

that

x, w1

S0,T0

(x),...,wk0−1

S0,T0

(x),wk0

S0,T0

(x)

are distinct for x ∈ A0. (Note here that ∀j k0,wj

S

0

,T0

(x) = wj

S0,T0

(x), ∀x ∈

A0.) Since δu(S0,S0) /2, we still have wn0

S ,T0

(x) = x for almost all

x ∈ A0.

Then, if k0 n, let 0 k0 k1 ≤ n be least such that on a set A1 ⊆ A0

of measure /4

x ∈ A1 ⇒ x, w1

S0,T0

(x),...,wk1−1

S0,T0

(x) are distinct and

wk1

S

0

,T0

(x) ∈ {x, w1

S

0

,T0

(x),...,wk1−1

S

0

,T0

(x)}

and repeat this process finitely many times to eventually find

¯

S,

¯

T with

δu(

¯

S, S0) , δu(

¯

T, T0) and a set

¯

A of positive measure such that

x ∈

¯

A ⇒ x, w1

¯

S,

¯

T

(x),...,wn

¯

S,

¯

T

(x) are distinct,

which is a contradiction, as for almost all x ∈

¯

A, wn

¯

S,

¯

T

(x) = x. ✷

A similar argument shows that the set of (S1,...,Sn) ∈ (APER ∩

[E])n

that generate a free group is also dense Gδ. So using also the Mycielski,

Kuratowski Theorem (see Kechris [Kec2], 19.1) this shows that there is a

Cantor set C ⊆ APER ∩ [E] generating a free group, so [E] contains a free

subgroup with continuum many generators.

(D) Note that the proof of 2.8 also shows the following.

Theorem 3.12 (Keane). The full group [E] is contractible for both the

weak and the uniform topologies.

Recently Kittrell and Tsankov [KiT] have shown that in fact ([E],u) is

homeomorphic to

2.

(E) Recall that E is a hyperfinite equivalence relation if it is induced by

an element of Aut(X, µ). In particular E0 is hyperfinite. The fundamen-

tal result about hyperfinite equivalence relations is the following classical

theorem of Dye. For a proof see, e.g., Kechris-Miller [KM].