26 I. MEASURE PRESERVING AUTOMORPHISMS

the free group Fm and En by a similar action of the free group Fn (1 ≤

m, n ≤ ∞), then if m = n, Em

∼

= En. Can one detect this by looking

at their full groups [Em], [En]? One can look for example at topological

generators. Recall that given a topological group G a subset of G is a

topological generator if it generates a dense subgroup of G. Denote by t(G)

the smallest cardinality of a set of topological generators of G. (Thus t(G) =

1 iff G is monothetic.) Clearly 1 ≤ t(G) ≤ ℵ0, if G is separable.

In the notation above, and keeping in mind that if Γ is dense in ([E],u),

then E = EΓ X , Gaboriau’s [Ga1] theory of costs immediately implies that

t([En]) ≥ n. In fact, Ben Miller pointed out that one actually has t([En]) ≥

n + 1. To see this, assume, towards a contradiction, that T1,...,Tn are

topological generators for [En] and let Γ be the group they generate. Since

En = EΓ

X

, and En has cost n, it follows from Gaboriau [Ga1], I.11, that

Γ acts freely. Let then S1,S2 be two distinct elements of Γ and (as in the

proof of 3.10) find disjoint Borel sets A1,A2 of positive measure such that

S1(A1),S2(A2) are also disjoint. Then there is S ∈ [En] with S|Ai = Si|Ai,

for i = 1, 2. Clearly, S cannot be in the uniform closure of Γ.

In particular, t([E∞]) = ℵ0. Note that, by 3.9, for any ergodic E we have

t([E]) ≥ 2. In an earlier version of this work, I have raised the question of

whether t([En]) ∞ (even in the case n = 1). This has now been answered

by the following result.

Theorem 4.12 (Kittrell-Tsankov [KiT]). An ergodic equivalence rela-

tion E is generated by an action of a finitely generated group iff t([E]) ∞

(where [E] is equipped with the uniform topology). Moreover if En is gen-

erated by a free, measure preserving, ergodic action of Fn, then t([En]) ≤

3(n + 1).

Kittrell and Tsankov [KiT] also proved that if n = 1, i.e., for ergodic

hyperfinite E, we have t([E]) ≤ 3. It is unknown whether in this case the

value of t([E]) is 2. For any n ∞, we have n + 1 ≤ t([En]) ≤ 3(n + 1).

It is unknown if t([En]) is independent of the action and, in case this has a

positive answer, what is the exact value of t([En]). In any case, the preceding

shows that t([En]) t([Em]), provided m+1 3(n+1). This appears to be

the first result providing a topological group distinction between [Em], [En],

provided m, n are suﬃciently far apart.

We remark that it is known that t(Aut(X, µ),w) = 2 and in fact the set

of pairs (g, h) ∈ Aut(X,

µ)2

that generate a dense subgroup is dense Gδ in

Aut(X,

µ)2

(see Grzaslewicz [Gr], Prasad [Pr], and Kechris-Rosendal [KR]

for a different approach). Also it is not hard to see that if U(H) is the

unitary group of an infinite dimensional Hilbert space, then t(U(H)) = 2,

and again, in fact, the set of pairs (g, h) ∈

U(H)2

that generate a dense

subgroup of U(H) is dense Gδ in

U(H)2.

This is because if U(n) is the

unitary group of the finite-dimensional Hilbert space

Cn

then (with some

canonical identifications) U(1) ⊆ U(2) ⊆ · · · ⊆ U(H) and

n

U(n) = U(H).

Then since each U(n) is compact and connected, it follows, e.g., by a result