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].
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