Definition 11.1 We say that a Borel equivalence relationE is finiteif every equivalence class is finite. We say thatE ishyperfiniteif there are finite Borel equivalence relationsE0⊆E1⊆. . .such thatE=SEn.
The equivalence relationE0is hyperfinite. LetFnbe the equivalence relation
onC
xFny⇔ ∀m > n x(m) =y(m).
ThenE0=SFn and eachFn is finite.
The main goal of this section will be to give the following characterizations of hyperfinite equivalence relations.
Theorem 11.2 LetE be a countable Borel equivalence relation. The following are equivalent:
i)E is hyperfinite;
ii) E is the orbit equivalence relation for a Borel action ofZ; iii)E≤BE0.
We first show that, for countable Borel equivalence relations, finite⇒tame ⇒hyperfinite.
Proposition 11.3 If E is a finite Borel equivalence relation, then E is tame.
Proof There is a Borel action of a countable groupG on X such that E is the orbit equivalence relation. Without loss of generality we may assume that X=Rso we can linearly orderX. Then
T ={x∈X :∀g∈G x≤gx} is a Borel transversal forE.
Proposition 11.4 If Eis tame countable Borel equivalence relation, thenE is hyperfinite.
Proof There is a countable group G such that E is the orbit equivalence relation onX. Suppose G={g0, g1, . . .}where g0=e. SinceE is tame, there is a Borel measurable selectors:X →X. Let
xEny⇔xEy and à x=y∨ à n ^ i=0 x=gis(x)∧ n ^ i=0 y=gis(x) !! . ThenxEns(x) if and only if x ∈ {gis(x) :i = 0, . . . , n}and ifx 6Ens(x) then
|[x]En|= 1. ThusEn is a finite equivalence relation andSEn=E.
Theorem 11.5 LetE be a countable Borel equivalence relation, thenE is hy- perfinite if and only if there are tame Borel equivalence relations E0 ⊆ E1 ⊆ E2⊆. . . withE=SnEn.
For a proof see [2] Theorem 5.1.
We mention a few important closure properties for hyperfinite equivalence relations.
Definition 11.6 If E is an equivalence relation on X we say that A⊆X is
fullforE if for all x∈X there isy∈Asuch thatxEy.
Proposition 11.7 i) If E⊆F andF is hyperfinite, thenE is hyperfinite. ii) IfE is hyperfinite andA⊆X is Borel, thenE|A is hyperfinite.
iii) If E is a countable Borel equivalence relation, A⊆X is Borel and full forE, andE|A is hyperfinite, thenA is hyperfinite.
iv) If E is a countable Borel equivalence relation, E ≤B E∗ and E∗ is
hyperfinite, thenE is hyperfinite.
Proof i) and ii) are obvious.
iii) SupposeE0⊆E1⊆E2⊆. . .are finite Borel equivalence relations on A such thatE|A=SEi. There is a countable groupG={g0, g1, . . . ,}such that
Eis the orbit equivalence relation for a Borel action ofGonX. Forx∈X, let nxbe least such thatgnxx∈A.
LetxFny if and only if
xEy∧(x=y∨(x≤n∧ny≤n∧gnxxFngnyy)).
ThenFn is a finite equivalence relation and SFn =E.
iv) Let f :X →Y be a Borel reductionE to a hyperfiniteE∗. Since E is
countable, the mapf has countable fibers. Thus by 7.21,B=f(X) is Borel and there is a Borel measurables:B →X such thatf(s(y)) =y for ally∈f(X). LetA = s(B) = {x ∈ X : s(f(x)) =x}. Then A is Borel and full in E. By ii) E∗|B is hyperfinite. But E|A is Borel isomorphic to E∗|B. By iii) E is
hyperfinite.
Z-actions
SupposeEis a Borel equivalence relation onX and<[x] is a linear order of [x]. We say that [x]7→<[x]is Borel if there is a BorelR⊆X×X×X such that
i)R(x, y, z)⇒(xEy∧xEz); ii)R(x, y, z)⇒y <[x]z;
iii) ifxEx1 thenR(x, y, z)⇔R(x1, y, z).
Theorem 11.8 LetE be a Borel equivalence relation onX. The following are equivalent:
i)E is hyperfinite;
ii) There is a Borel [x]7→<[x] such that each infiniteE-class has order type
iii) There is a Borel [x]7→<[x]such that each infiniteE-class has order type
Z.
iv) There is a Borel action of Zon X such that E is the orbit equivalence relation.
v) There is a Borel automorphism T : X → X such that E-equivalence classes areT-orbits.
Proof
It is clear that iv)⇔v)
i)⇒ ii) Let E0 ⊆E1 ⊆E2 ⊆. . . be finite Borel equivalence relations such thatE=SEn. We may assume thatE0 is equality. We may also assume that
there is an ordering<ofX.
We inductively define<[x]En as follows.
1)<[x]E0 is trivial, since [x]E0 ={x}.
2) Supposey, zEnxandyEn−1z, theny <[x]En z if and only ify <[y]En−1 z. 3) Suppose y, zEnx and y 6En−1z. Let ybbe the <[y]En−1-least element of [y]n−1andbzbe the<[z]En−1-least element of [z]En−1. Ifby <z, thenb y <[x]En z.
Otherwisez <[x]En y.
In other words: we order [x]Enbe breaking it into finitely manyEn−1classes C1, . . . , Cm. We then order the classes Ci by letting yi be the <[yi]En−1-least element and saying thatCi< Cj ifyi< yj.
Let <[x]E= S <[
x]En. If xEny andx <[x] z <[x] y, thenxEnx. It follows
that<[x]is a discrete union of finite orders. Thus<[x]is either a finite order or has order typeω,ω∗ orZ.
We need only argue that the assignments [x] 7→<[x]En is Borel. The only
difficulty is picking bx the<[x]En-least element of [x]En. There is a countable
groupsGand a Borel actions ofGonX such that En is the orbit equivalence
relation ofGn. Then
y=xb⇔(yEnx∧ ∀g∈Gn y≤[x]En gx).
This is easily seen to be Borel.
ii) ⇒ iii) We may assume that E is the orbit equivalence relation for the action of a countable groupG. Since
{x:∃g∈G∀h∈G hx≤[x]gx} and
{x:∃g∈G∀h∈G gx≤[x] hx}
are Borel we can determine the order type of each class. If a class has order typeω ofω∗ we can reorder it so that it has order typeZ. For example if [x] is x0<[x]x1<[x]< . . . we define a new order<∗ so that
. . . x5<∗x3<∗x1<∗x0<∗x2<∗x4< . . . Theω∗ case is similar. This can clearly be done in a Borel way.
iii) ⇒iv) We define a Borel automorphism g :X →X such thatE is the orbits of g. If x is the <[x]-maximal element of [x], then [x] is the <[x]-least
element of [x]. Otherwise letg(x) be the<[x]-successor ofx. Arguing as above
gis Borel. We letZact onX bynx=g(n)x. ClearlyEis the orbit equivalence relation.
iv)⇒iii) Letg:X →X be a Borel automorphism such thatE-classes are g-orbits. Then X0 ={x :∃n 6= 0 :g(n)x=x}is Borel. On X0 we can define <[x] using <is a fixed linear order of X. Thus, without loss of generality, we may assume that everyE-class is infinite. But then we can define x <[x] y if and only if there is an n >0 such thatg(n)x=y. Clearly this is aZ-ordering of [x].
iii) ⇒i) We may assume that E is an equivalence relation onC. For each equivalence classCwe define a treeTC⊆2<ω by
TC={σ∈2<ω:∃x∈C x⊃σ}.
There is a Borel automorphismgsuch thatE-classes areg-orbits. Since T[x] ={σ:∃n∈Z:g(n)x⊃σ},
the functionx7→T[x] is Borel measurable. ClearlyTCis infinite. LetzC∈[TC]
be the leftmost path inTC.
Claim The functionsx7→z[x] is Borel measurable. We defineσx
0 ⊂σx1 ⊂. . .such that{τ ∈T[x]:τ ⊇σxi}is infinite. Letσx0 =∅ and σx
i+1 = σixbj where j is least such that {τ ∈ T[x] : τ ⊇ σixbj} is infinite.
Thenz[x]=Sσix. It is easy to see that (T[x], z[x]) is Π01. Thusx7→z[x] is Borel measurable.
There are several cases to consider. It will be clear that deciding which case we are in is Borel.
case 1: zC∈C.
Forx∈Cwe definexEnyif and only ifx=yor there arei, jwith|i|,|j| ≤n
such thatx=g(i)z[
x] andy=g(j)z[x].
Form∈NletCm={x∈C:x|m=zC|m}.
case 2: There is anmsuch that Cm has a<C-least element.
Letmbe least such thatCm has a least elementwC. Forx∈C, we define
xEny if and only ifx=yor there arei, j with|i|,|j| ≤nsuch thatx=g(i)w[x] andy=g(j)w[
x].
case 3: There is anmsuch that Cm has a<C-maximal element.
Similar. case 4: Otherwise.
We haveC0⊇C1 ⊇C2 ⊇. . .. Since we are not in case 1,TCi =∅. Since
we are not in case 2 or 3,Ci has no smallest or largest element.
We defineEn onCby: xEny if and only if (x∈Cn andx=y) or and there
isi >0 such thatg(i)x=yand g(
Clearly eachEn class is finite and ifxEy, thenxEnyfor all sufficiently large
n.
The i)⇔iii) is due to Slaman and Steel. The direction iv) ⇒ i) is due to Weiss.
It follows immediately that there is a universal hyperfinite Borel equivalence.
Corollary 11.9 If E is a hyperfinite Borel equivalence relation, then
E≤B E(Z,C).
Recall that an action of GonX is a free action if gx6=hxfor any x∈X andg6=h. Our proof shows the following.
Corollary 11.10 IfE is a hyperfinite equivalence relation on a standard Borel space X and every E class if infinite, then E is the orbit equivalence relation for a free Borel action ofZ onX.