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Cardinality and Equivalence Relations
Greg Hjorth
University of Melbourne
Tarski Lectures
University of California, Berkeley
April 5, 2010
1
1 The broad outline of these talks
1. The basic idea of cardinality. (At the begin-ning at least, make very few mathematicalassumptions of the audience.)
2. Slight modifications of this concept can leadto a spectrum of notions which resemble thenotion of size or cardinality. (“Borel cardi-nality” or “e!ective cardinality”, or even car-dinality in L(R)).
3. Some of these notions are implicit in mathe-matical activity outside set theory. (For in-stance the work on dual of a group by GeorgeW. Mackey, or scattered references to e!ec-tive cardinality in the writing of Alain Connes.)
4. In the 80’s set theorists such as Harvey Fried-man, Alexander Kechris, among others, be-gan to suggest a way to explicate this ideaaround the concept of Borel reducibility
2
5. The outpouring of activity in this area overthe last 15 -20 years. Su Gao, AlexanderKechris, Alain Louveau, Slawek Solecki, Si-mon Thomas, Boban Velickovic, among manyothers.
6. Dichotomy theorems for Borel equivalence re-lations (for instance, the Harrington, Kechris,and Louveau extension of Glimm-E!ros).
7. Specific classification problems in mathemat-ics (for instance the finite rank torsion freeabelian groups)
8. Dynamical methods to analyze equivalencerelations in the absence of reasonable dichotomytheorems (for instance turbulence)
9. Interactions between the theory of countableequivalence relations and the theory of orbitequivalence
3
2 Equivalence relations and invariants
Definition Let X be a set. E ! X " X issaid to be an equivalence relation if
1. it is reflexive (xEx all x # X)
2. symmetric (xEy implies yEx)
3. transitive (xEy along with yEz implies xEz).
Example 1. Let X be the set of people. LetE be the equivalence relation of having thesame height.
2. Let X again be the set of all people. LetE be the equivalence relation of having thesame mother.
3. Let X be the set of all planets. Let E be theequivalence relation of being located in thesame universe.
4
Definition For E an equivalence relation ona set X , a complete invariant or classificationof E is a “reasonable” or “explicit” or “natural”function
f : X $ I
such that for all x1, x2 # X we have
x1Ex2
if and only if
f (x1) = f (x2).
This really a pseudo-definition, held completelyhostage to how we best make sense of “reason-able” or “explicit”.
Part of the story are the attempts to make thisidea more precise, and this in turn connects inwith variations on the concept of cardinality
5
Example 1. For E the equivalence relation ofhaving the same height we clearly do have anatural classification. Namely: Height mea-sured in feet and inches. Assign to each x inthe set of people the height measured in feetand inches as f (x).
2. For E the equivalence relation of having thesame mother, it would likewise seem that thereis a very explicit invariant: Assign to eachperson x their mother as f (x).
3. For E being the equivalence relation of beingin the same universe, the situation is not soclear.
We could try to for instance assign to eachplanet the actual universe in which they live,but it is not clear that this is doing muchmore than assigning to each x the entire setof all y for which xEy.
6
3 Cardinality
Definition A function
f : X $ Y
is an injection if for all x1, x2 # X ,
x1 %= x2
impliesf (x1) %= f (x2).
In other words, an injection is a function whichsends distinct points to distinct images.
7
Example 1. Let X be the set of people and letY be the set of all human heads. Let
f : X $ Y
assign to each x its head. This is presumably(barring some very unusual case of siamesetwins) an injection.
2. Let X be the set of all people who have everlived and let
f : X $ X
assign to each X its mother. (Here we areignoring minor chicken-egg questions aboutthe first ever mother). This is clearly not aninjection. There are cases of distinct peoplehaving the same mother.
8
Definition A function
f : X $ Y
is a surjection if for each y # Y there is somex # X with
f (x) = y.
A function which is both injective and surjectiveis called a bijection.
In rough terms, a bijection between X and Yis a way of marrying all the X ’s o! with allthe Y ’s with no unmarried Y ’s left over. (Hereassuming no polygamy allowed).
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Definition Given two sets X and Y , we saythat the cardinality of X is less than the car-dinality of Y , written
|X| & |Y |,
if there is an injection
f : X $ Y.
Theorem 3.1 (Schroeder-Bernstein) If
|X| & |Y |
and|Y | & |X|,
then there is a bijection between the two sets.
Intuitively not so outrageous. If we say thata set has size 4, and count of the elements 1,2, 3, 4, we are implicitly placing that set in abijection with the set {1, 2, 3, 4}.
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4 The axiom of choice
Definition A set ! is an ordinal if:
1. it is transitive – " # ! along with # # "implies # # !; and
2. it is linearly ordered by # – if ", # are bothin !, then
" # #,
or# # ",
or# = ".
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', the set having no members, which set theo-rists customarily identify with 0.
1 = {0}, the set whose only member is 0.2 = {0, 1}, gives the usual set theoretical def-
inition of 2. Then we keep going with 3 ={0, 1, 2}, 4 = {0, 1, 2, 3}, and so on.
We reach the first infinite ordinal with the setof natural numbers:
$ = {0, 1, 2, ...}.
This again leads to a whole new ladder of ordi-nals:
$ + 1 = {0, 1, 2, ...,$},
$ + 2 = {0, 1, 2, ...,$,$ + 1},
$ + 3 = {0, 1, 2, ...,$,$ + 1,$ + 2},
and onwards:
$ + $ = {0, 1, 2, ...,$,$ + 1,$ + 2, ...},
$+$+1 = {0, 1, 2, ...,$,$+1,$+2, ...,$+$},
$"$ = {0, 1, 2, ...,$,$+1, ...$+$, ...$+$+$, ...},
ad infinitum.12
Lemma 4.1 The ordinals themselves are lin-early ordered: If ", # are both ordinals, then
" # #,
or# # ",
or# = ".
The Axiom of Choice: (In e!ect) Every setcan be placed in a bijection with some ordinal.
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Definition An ordinal is said to be a cardinalif it cannot be placed in a bijection with anysmaller ordinal.
So for instance, $ (or (0 as it is sometimescalled in this context) is indeed a cardinal: Thesmaller ordinals are finite.
But not $ + $: Define
f : $ + $ $ $,
n )$ 2n,
$ + n )$ 2n + 1.
14
The consequences of all this for the theory ofcardinality:
Every set can be placed in a bijectionwith an ordinal.
The cardinals are a linearly ordered set.
This is parallel to a form of utilitarianism:There is only one good (human happiness) andthat good can be compared in order and amount.
There is only one notion of “size” and the car-dinals can be compared in order.
15
5 Variations on the notion of cardinality
Recall that we set |X| & |Y | if there is an in-jection from X to Y .
However, it does make sense to look at paral-lel definitions for classes of injections which aremore narrow than simply the class of all injec-tions.
Fix " some class of functions (for instance, allBorel functions, all functions in L(R)).
We might want to say that the "-cardinalityof X is less than equal to that of Y if there issome injection f # " with
f : X $ Y.
16
This should be compared to the problem ofclassifying an equivalence relation.
Definition For E and equivalence relation onX , and x # X , let [x]E be the set of all y # Xwith
xEy.
Then let X/E be the set of all equivalence classes:
{[x]E : x # X}.
For instance, if X is the set of all people, andE is the equivalence relation of having the sameheight, then
[Greg]Ewould be the set of all people who are 5’6” tall.
X/E would consist of the set of all “group-ings” of people where they are categorized strictlyby height.
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Then a classification of E would in some formbe an “appropriate” function
f : X $ I
which induces an injection
f : X/E $ I
via lettingf (C)
take the valuef (y)
for any y in the equivalence class C.
The assumption xEy * f (x) = f (y) ensuresf is well defined.
The assumption f (x) = f (y) * xEy ensuresf is an injection.
However it remains to resolve the definition ofwhat we should count as an “appropriate” orreasonable class of possible functions f .
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6 The ideas of G. W. Mackey
One particular approach to theory of what shouldcount as reasonable functions for the point ofview of classification has been suggested by workof Mackey on group representations dating backto the middle of the last century.
The story now becomes considerably more math-ematical. I will start first by describing theproblem Mackey considered, and only then theexplication of reasonable his work suggests.
Definition For H a Hilbert space, U(H) de-notes the group of unitary operators on H .That is to say, the set of all linear bijections
T : H $ H
such that for all u, v # H
+T (v), T (u), = +u, v,.
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Definition For G a countable group, a unitaryrepresentation of G (on the Hilbert space H)is a homomorphism
% : G $ U(H)
g )$ %g.
We then say that a representation is irreducibleif the only closed subspaces of H which are in-variant under
{%g : g # G}
are the trivial ones: 0 and H .
Let Irr(G,H) denote the collection of irreducibleunitary representations of G on H .
20
Definition Two unitary representations
% : G $ U(H%)
& : G $ U(H&)
are equivalent, written
% -= &,
if they are unitarily conjugate in the sense that
there is a unitary isomorphism
T : H% $ H&
such that at every g # G
%g = T . &g . T/1.
21
Theorem 6.1 If % : Z $ U(H) is an irre-ducible representation, then:
1. H is one dimensional; and
2. there is some z # C such that at every' # Z we have
%'(v) = z' · v
all v # H; and
3. two distinct irreducible representations areequivalent if and only if they have the samez # C associated to them.
Thus we obtain a complete classification of ir-reducible representations of Z by their associ-ated z # C. It is like we can view Irr(Z, H)/ -=as a subset of C.
It turns out that a similar, though somewhatmore complicated, classification can be given forany abelian group.
22
In broad terms Mackey was led to ask: Forwhich groups G can we reasonably classify thecollection of equivalence classes
Irr(G, H)/ -=
by points in some concrete space such as C?
Before even groping towards an answer, onemight first want to make the question precise.
Mackey did make the question precise, but thisin turn requires the introduction of ideas lyingat the foundations of descriptive set theory.
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7 Polish spaces and Borel sets
Definition A topological space is said to bePolish if it is separable and it admits a completecompatible metric.
We then say that the Borel sets are those ap-pearing in the smallest (-algebra containing theopen sets.
A set X equipped with a (-algebra is said to bea standard Borel space if there is some choiceof a Polish topology giving rise to that (-algebraas its collection of Borel sets.
A function between two Polish spaces,
f : X $ Y,
is said to be Borel if for any Borel B ! Y thepullback f/1[B] is Borel.
24
Some examples
1. Any separable Hilbert space is Polish.
2. If H is a separable Hilbert space, then U(H)is a closed subgroup of its isometry group andhence Polish.
3. If H is a separable Hilbert space and G is acountable group, then
!
G
U(H)
is a countable product of Polish spaces andhence Polish.
4. Then the collection of unitary representationsof G is a closed subspace of
"G U(H), and
hence Polish.
5. Finally it is a slightly non-trivial fact thatthe collection of irreducible representations isa G) subset of the collection of all represen-tations, and hence Polish.
25
Definition (Mackey) An equivalence relationE on a Polish space X is smooth if there is aanother Polish space Y and a Borel function
f : X $ Y
such that for all x1, x2 # X we have
f (x1) = f (x2)
if and only ifx1Ex2.
A countable group G has smooth dual if forany separable Hilbert space H , the equivalencerelation -= on Irr(G,H) is smooth.
Question (Mackey, in e!ect) Which groupshave smooth dual?
26
There are various answers in the literature toMackey’s question, including Glimm’s solutionof the Mackey conjecture, which applies notjust to discrete groups but more generally lcsctopological groups.
In the case of discrete groups there is a com-pletely algebraic characterization.
Theorem 7.1 (Thoma) A countable group Ghas smooth dual if and only if it has an abeliansubgroup with finite index.
27
There are some very simple, from the point ofview of Borel complexity, non-smooth equiva-lence relations.
Definition Equip
2N =df
!
N
{0, 1}
with the product topology.
Let E0 be the equivalence relation of eventualagreement on 2N.
Lemma 7.2 E0 is not smooth.
E0 itself is F( as a subset of 2N. The com-plexity of its classification problem has little dowith any complexity it might have as a subsetof 2N " 2N.
28
It turns out that at the base of Glimm’s proofof the Mackey conjecture is a theorem to thee!ect that under certain circumstances E0 is thecanonical obstruction to smoothness.
This was generalized by Ed E!ros.
The final and ultimate generalization to theabstract theory of Borel equivalence relationswas obtained by Leo Harrington, Alexander Kechris,and Alain Louveau in the late 1980’s and in turnsparked a new direction of research in descrip-tive set theory.
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Classification Problems in Mathematics
Greg Hjorth
University of Melbourne
Tarski Lectures
University of California, Berkeley
April 7, 2010
1
1 Recall
Definition The cardinality of X is less thanor equal to Y ,
|X| ! |Y |,
if there is an injection from X to Y .
This might suggest a notion of “cardinality”where we restrict our attention to some restrictedclass of injections.
This in turn could relate to the idea that anequivalence relation E on a set X is in somesense classifiable if there is a “reasonably nice”or “natural” or “explicit” function
f : X " I
which induces (via x1Ex2 # f (x1) = f (x2))an injection
f : X/E " I.
2
In the context of unitary group representationa definition exactly along these lines was pro-posed by G. W. Mackey.
Definition (Mackey) An equivalence relationE on a Polish space X is smooth if there is aPolish space Y and a Borel function f : X ! Ysuch that
x1Ex2 " f (x1) = f (x2).
In the way of context and background
1. Borel functions are considered by many math-ematicians to be basic and uncontroversial,and concrete in a way that a function sum-moned in to existence by appeal to the axiomof choice would not.
2. Many classification problems can be cast inthe form of understanding an equivalence re-lation on a Polish space
3
2 The entry of descriptive set theorists
In the late 80’s two pivotal papers suggested avariation and generalization of Mackey’s defini-tion.
A Borel Reducibility Theory for Classes ofCountable Structures, H. Friedman and L. Stan-ley, The Journal of Symbolic Logic, Vol.54, No. 3 (Sep., 1989), pp. 894-914
A Glimm-E!ros Dichotomy for Borel Equiv-alence Relations, L. A. Harrington, A. S. Kechrisand A. Louveau, Journal of the American
Mathematical Society, Vol. 3, No. 4 (Oct.,1990), pp. 903-928
Neither paper referenced the other, and yetthey used the exact same terminology and no-tation to introduce a new concept.
4
Definition Given equivalence relations E andF on X and Y we say that E is Borel reducibleto F , written
E !B F,
if there is a Borel function
f : X " Y
such that
x1Ex2 # f (x1)Ff (x2).
In other words, the Borel function f inducesan injection
f : X/E " Y/F.
The perspective of Friedman and Stanley wasto compare various classes of countable struc-tures under the ordering !B. The Harrington,Kechris, Louveau paper instead generalized ear-lier work of Glimm and E!ros in foundationalissues involving the theory of unitary group rep-resentations.
5
Definition Let E0 be the equivalence relationof eventual agreement on 2N. For X a Polishspace let id(X) be the equivalence relation ofequality on X .
Thus in the above notation we can recast Mackey’sdefinition of smooth: An equivalence relation Eis smooth if for some Polish X we have
E !B id(X).
It turns out that for any uncountable Polishspace X we have
id(R) !B id(X)
andid(X) !B id(R).
6
Definition An equivalence relation E on Pol-ish X is Borel if it is Borel as a subset of X!X .
Theorem 2.1 (Harrington, Kechris, Louveau)Let E be a Borel equivalence relation. Thenexactly one of the following two conditionsholds:
1. E "B id(R);
2. E0 "B E.
Moreover 1. is equivalent to E being smooth.
This breakthrough result, this archetypal di-chotomy theorem, suggested the possibility ofunderstanding the structure of the Borel equiv-alence relations up to Borel reducibility, whichin turn has become a major project in the lasttwenty years, which I will survey on Friday.
7
3 Examples of Borel reducibility in mathematical practice
It turns out that many classical, or near classi-cal, theorems can be recast in the language ofBorel reducibility.
Example Let (X, d) be a complete, separable,metric space. Let K(X) be the compact subsetsof X – equipped with the metric D(K1,K2)equals
supx!K1d(x,K2) + supx!K2
d(x,K1),
where d(x,K) = infz!Kd(x, z).
Let E be the equivalence relation of isome-try on K(X). Then Gromov showed that E issmooth. In other words,
E "B id(R).
8
Example Let H be a separable Hilbert spaceand U(H) the group of unitary operators of H .
Let != be the equivalence relation of conjugacyon U(H), which is in e!ect the isomorphismrelation considered in the last talk: T1
!= T2 if
"S # U(H)(S $ T1 $ S%1 = T2).
1. In the case that H is finite dimensional, everyT # U(H) can be diagonalized. This gives areduction of != to the equality of finite subsetsof C, and hence a proof that != is smooth.
2. In the case that H is infinite dimensional,the situation is considerably more subtle, butthe spectral theorem allows us to write eachelement of U(H) as a kind of direct integralof rotations.
9
Definition Let S1 be the circle:
{z ! C : |z| = 1}
in the obvious, and compact, topology. LetP (S1) be the collection of probability mea-sures on S1 – this forms a Polish space in thetopology it inherits from being a closed sub-set C(S1)" in the weak star topology (via theRiesz representation theorem). For µ, ! !P (S1), set µ # ! if they have the same nullsets.
It then follows from the spectral theorem that#=$B# .
The spectral theorem is often considered, thoughwithout the use of the language of Borel re-ducibility, to provide a classification of theinfinite dimensional unitary operators up toconjugacy.
10
Example For S a countable set, may identifyP(S) with
2S =!
S
{0, 1}
and thus view it as a compact Polish space inthe product topology.
A torsion free abelian (TFA) group A is saidto be of rank ! n if there are a1, a2, ...an " Asuch that every b " A has some m " N withm · b " #a1, ..., an$.
Up to isomorphism, the rank ! n TFA groupsare exactly the subgroups of (Qn, +), and thusform a Polish space as a subset of P(Qn).
Let %=n be the isomorphism relation on sub-groups of (Qn, +). In the language of Borelreducibility a celebrated classification theoremcan be rephrased as:
Theorem 3.1 (Baer) %=1!B E0.
11
Example Let Hom+([0, 1]) be the orientationpreserving homeomorphisms of the closed unitinterval. In the sup norm metric, this forms aPolish space.
Let !=Hom+([0,1]) be the equivalence relation ofconjugacy.
There is a kind of folklore observation to thee!ect that every element of Hom+([0, 1]) can beclassified symbolically, by recording the max-imal open intervals on which it is increasing,decreasing, or the identity.
This translates into classifying
Hom+([0, 1])/ !=Hom+([0,1])
by countable linear orderings with equipped withunary predicates Pinc and Pdec up to isomor-phism. Those in turn can be viewed as forminga closed subset of 2N"N " 2N " 2N, and weobtain
!=Hom+([0,1])#B!=2N"N"2N"2N .
12
4 The space of countable models
Definition LetL be a countable language. ThenMod(L) is the set of all L structures with un-derlying set N.
Definition Let !qf be the topology with basicopen sets of the form
{M ! Mod(L) : M |= "(#a)}
where "(#x) is quantifier free and #a ! N<".
!fo is defined in a parallel fashion, except with"(#x) ranging over first order formulas, and moregenerally for F # L$1,$ a countable fragmentwe define !F similarly with "(#x) ! F .
It is not much more than processing the defi-nitions to show !qf is Polish. For instance for Lconsisting of a single binary relation, we obtaina natural isomorphism with 2N$N. It can beshown, however, that the others are Polish, andall these examples have the same Borel struc-ture.
13
Definition For a sentence ! ! L"1," we let"=! be isomorphism on Mod(!), the set of M !Mod(L) with M |= !.
"=! is universal for countable structures ifgiven any countable language L# we have
"=Mod(L#)$B"=! .
Theorem 4.1 (Friedman, Stanley) The fol-lowing are universal for countable structures:Isomorphism of countable trees, countable fields,and countable linear orderings. Isomorphismof countable torsion abelian groups is not uni-versal for countable structures.
One tends to obtain universality1 for such aclass of countable structures , except when thereis an “obvious” reason why this must fail.
For instance, if the isomorphism relation is “es-sentially countable”.
1The major being torsion abelian groups. The case for torsion free abelian groups remains puzzlingly open, despite
strong indicators it should be universal
14
5 Essentially countable equivalence relations
Definition A Borel equivalence relation F onPolish Y is countable if every equivalence classis countable.
An equivalence relation E on a Polish space isessentially countable if it is Borel reducible toa countable equivalence relation.
An equivalence relation E is universal for es-sentially countable if it is essentially countableand for any other countable Borel equivalenceF we have F !B E.
Theorem 5.1 (Jackson, Kechris, Louveau)Universal essentially countable equivalence re-lations exist. In fact, for F2 the free group ontwo generators, the orbit equivalence relationof F2 on 2F2 is essentially countable.
15
Fact 5.2 If an equivalence relation E is es-sentially countable, then for some2 countablelanguages L we have E !B
"=Mod(L).
Theorem 5.3 (Kechris) If G is a locally com-pact Polish group acting in a Borel manneron a Polish space X, then the resulting equiv-alence relation is essentially countable.
In the context of isomorphism types of classesof countable structures, one can characterize whenan equivalence relation is essentially countablein model theoretic terms.
Roughly speaking a class of countable struc-tures with an appropriately “finite character”will be essentially countable.
In particular, if M # Mod(L) satisfying ! isfinitely generated, then "=! is essentially count-able.
2In fact, “most”16
Theorem 5.4 (Thomas, Velickovic) Isomor-phism of finitely generated groups is univer-sal for essentially countable.
As in the case of general countable structures,the tendency is for classes of essentially count-able structures to be universal unless there issome relatively obvious obstruction.
In a paper with Kechris, we made a ratherarrogant, reckless, and totally unsubstantiated,conjecture that isomorphism for rank two tor-sion free abelian groups would be universal foressentially countable.
Since E0 is not universal for essentially count-able, this was hoped to explain the inability ofabelian group theorists to find a satisfactoryclassification for the higher finite rank torsionfree abelian groups.
17
6 The saga of finite rank torsion free abelian groups
Although many well known mathematical clas-sification theorems have a direct consequence forthe theory of Borel reducibility, a major moti-vation has been to use the theory of Borel re-ducibility to explicate basic obstructions to theclassification of certain classes of isomorphism.
One of the most clear cut cases has been thesituation with finite rank torsion free abeliangroups.
Recall that !=n is being used to describe theisomorphism relation on rank " n torsion freeabelian groups, where we provide a model of thefull set of isomorphism types by considering thesubgroups of Qn.
Theorem 6.1 (Baer, implicitly, 1937)!=1"B E0.
18
A kind of mathematically precise justificationfor the vague feeling that rank two torsion freeabelian groups did not admit a similar classifi-cation was provided by:
Theorem 6.2 (Hjorth, 1998) !=2 is not Borelreducible to E0.
In some sense this addressed the soft philo-sophical motivation behind the conjecture withKechris, but not the hard mathematical formu-lation with which it faced the world. This wasleft to Simon Thomas, who in a technically bril-liant sequence of papers showed:
Theorem 6.3 (Thomas, 2002, 2004) At ev-ery n
!=n<B!=n+1 .
19
In general, and this lies at the heart of thetechnical mountains Thomas had to overcome,almost all the results to show that one essen-tially countable equivalence relation is not Borelreducible to another rely on techniques comingentirely outside logic, such as geometric grouptheory, von Neumann algebras, and the rigiditytheory one finds in the work of Margulis andZimmer.
In recent years the work of logicians in thisarea has begun to communicate and interactwith mathematicians in quite diverse fields.
However, it has gradually become clear thatmany of the problems we would most dearlylike to solve will not be solvable by the measuretheoretic based techniques being used in theseother fields. For instance....
20
Question Let G be a countable nilpotent groupacting in a Borel manner on a Polish space withinduced orbit equivalence relation EG. Must wehave
EG !B E0?
The problem here is that with respect to anymeasure we will have EG !B E0 on some conullset, and thus measure will not be a suitablemethod for proving the existence of a counterex-ample. In an enormously challenging and strik-ingly original fifty page manuscript, Su Gao andSteve Jackson showed EG !B E0 when G isabelian.
Many of these issues relate to open problemsin the theory of Borel dichotomy theorems andthe global structure of the Borel equivalence re-lations under !B.3
3Next talk21
7 Classification by countable structures
Definition An equivalence relation E on a Pol-ish space X is classifiable by countable struc-tures if there is a countable language L and aBorel function
f : X ! Mod(L)
such that for all x1, x2 " X
x1Ex2 # f (x1) $= f (x2).
Here one might compare algebraic topology,where algebraic objects considered up to iso-morphism are assigned as invariants for classesof topological spaces considered up to homeo-morphism.
Again it turns out that some well known classi-fication theorems have the direct consequence ofshowing that some naturally occurring equiva-lence relation admits classification by countablestructures.
22
Example Recall !=Hom+([0,1]) as the isomor-phism relation on orientation measure preserv-ing transformations of the closed unit interval.Then the folklore observation mentioned frombefore in particular shows that !=Hom+([0,1]) isclassifiable by countable structures.
Example A Stone space is a compact zero di-mensional Hausdor! space. There is a fixedtopological space X (for instance, the Hilbertcube), such that every separable Stone spacecan be realized as a compact subspace of X .Then S(X), the set of all such subspaces, formsa standard Borel space, and we can let !=S(X)be the homeomorphism relation on elements ofS(X).
Stone duality, the classification of Stone spacesby their associated Boolean algebras, in particu-lar shows that !=S(X) is classifiable by countablestructures.
23
Example For ! the Lebesgue measure on [0, 1],let M! be the group of measure preservingtransformations of ([0, 1]!) (considered up toequality a.e.).
A measure preserving transformation T is saidto be discrete spectrum if L2([0, 1],!) is spannedby eigenvalues for the induced unitary operator
UT : f "# f $ T%1.
It follows from the work of Halmos and vonNeumann that such transformations consideredup to conjugacy in M! are classifiable by count-able structures.
24
Example The search for complete algebraic in-variants has recurrent theme in the study of C!-algebras and topological dynamics.
Consider minimal (no non-trivial closed invari-ant sets) homeomorphisms of 2N
Let "C(2N) be conjugacy of orbit equivalencerelations: Thus
f1 "C(2N) f2
if there is some homeomorphism g conjugatingtheir orbits:
#!x(g[{f "1(!x) : " $ Z}] = {f "
2(g(!x)) : " $ Z}).
Giordano, Putnam, and Skau produce count-able ordered abelian groups which, consideredup to isomorphism, act as complete invariants.
Their theorem implicitly shows "C(2N) to beclassifiable by countable structures.
25
8 Turbulence
This a theory, or rather a body of techniques,explicitly fashioned to show when equivalencerelations are not classifiable by countable struc-tures.
Definition Let G be a Polish group acting con-tinuously on a Polish space X . For V an openneighborhood of 1G, U an open set containingx, we let
O(x, U, V ),
the U-V -local orbit, be the set of all x ! [x]Gsuch that there is a finite sequence
(xi)i"k # U
such thatx0 = x, xk = x,
and eachxi+1 ! V · xi.
26
Definition Let G be a Polish group acting con-tinuously on a Polish space X . The action issaid to be turbulent if:
1. every orbit is dense; and
2. every orbit is meager; and
3. for x ! X , the local orbits of x are all some-where dense; that is to say, if V is an openneighborhood of 1G, U is an open set contain-ing x, then closure of O(x, U, V ) contains anopen set.
Theorem 8.1 (Hjorth) Let G be a Polish groupacting continuously on a Polish space X withinduced orbit equivalence relation EG.
If G acts turbulently on X, then EG is notclassifiable by countable structures.
This has been the engine behind a number ofanti-classification theorems.
27
Example (Kechris, Sofronidis) Infinite dimen-sional unitary operators considered up to uni-tary conjugacy do not admit classification bycountable structures.
Example (Hjorth) The homeomorphism groupof the unit square,
Hom([0, 1]2),
considered up to homeomorphism does not ad-mit classification by countable structures.
Example (Gao) Countable metric spaces up tohomeomorphism does not admit classificationby countable structures.
Example (Tornquist) Measure preserving ac-tions of F2 up to orbit equivalence do not admitclassification by countable structures.
28
Borel Equivalence Relations:Dichotomy Theorems and Structure
Greg Hjorth
University of Melbourne
Tarski Lectures
University of California, Berkeley
April 9, 2010
1
1 The classical theory of Borel sets
Definition A space is Polish if it is separableand admits a complete metric.
We then say that the Borel sets are those ap-pearing in the smallest !-algebra containing theopen sets.
A set X equipped with a !-algebra is said to bea standard Borel space if there is some choiceof a Polish topology giving rise to that !-algebraas its collection of Borel sets.
A function between two Polish spaces,
f : X ! Y,
is said to be Borel if for any Borel B " Y thepullback f#1[B] is Borel.
We have gone through a number of examplesin the first two talks. There is a sense in whichPolish spaces are ubiquitous.
2
The notion of standard Borel space is slightlymore subtle.
However it turns out that there are many ex-amples of standard Borel spaces which possessa canonical Borel structure, but no canonicalPolish topology.1
Theorem 1.1 (Classical) If X is a Polishspace and B ! X is a Borel set, then B(equipped in the !-algebra of Borel subsetsfrom the point of view of X) is standard Borel.
Theorem 1.2 (Classical; the “perfect set the-orem”) If X is a Polish space and B ! X isa Borel set, then exactly one of:
1. B is countable; or
2. B contains a homeomorphic copy of Can-tor space, 2N (and hence has size 2"0).
1Indeed, since we are mostly only considering Polish spaces up to questions of Borel structure, it is natural to discount thespecifics of the Polish topology involved.
3
Theorem 1.3 (Classical) If X is a standardBorel space, then the cardinality of X is oneof {1, 2, 3, ....,!0, 2
!0}.
Moreover!
Theorem 1.4 (Classical) Any two standardBorel spaces of the same cardinality are Borelisomorphic.
Here we say that X and Y are Borel isomor-phic if there is a Borel bijection
f : X " Y
whose inverse is Borel.2
Thus, as sets equipped with their !-algebrasthey are isomorphic.
There is a similar theorem for quotients of theform X/E, E a Borel equivalence relation.
2In fact it is a classical theorem that any Borel bijection must have a Borel inverse4
2 The analogues for Borel equivalence relations
Definition If X is a standard Borel space, anequivalence relation E on X is Borel if it ap-pears in the !-algebra on X ! X generated bythe rectangles A!B for A and B Borel subsetsof X .
Theorem 2.1 (Silver, 1980) Let X is a stan-dard Borel space and assume E is a Borelequivalence relation on X. Then the cardi-nality of X/E is one of
{1, 2, 3, ....,"0, 2"0}.
However here there is no moreover.
In terms of Borel structure, and the situa-tion when X/E is uncountable, there are vastlymany possibilities at the level of Borel structure.
5
Definition Given equivalence relations E andF on standard Borel X and Y we say that E isBorel reducible to F , written
E !B F,
if there is a Borel function
f : X " Y
such that
x1Ex2 # f (x1)Ff (x2).
We say that the Borel cardinality X/E isless than the Borel cardinality of Y/F , writ-ten
E <B F,
if there is a Borel reduction of E to F but noBorel reduction of F to E.
In the language Borel reducibility, there is asharper version of Silver’s theorem, which healso proved without describing it in these terms.
6
Theorem 2.2 (Silver) Let E be a Borel equiv-alence relation on a standard Borel space.Then exactly one of:
1. E !B id(N); or
2. id(R) !B E.
One of the major events in the prehistory ofthe subject is Leo Harrington’s alternate and farshorter proof of Silver’s result using a technol-ogy called Gandy-Harrington forcing.
Building on this technology with the combina-torics of an earlier argument due to Ed E!ros,the whole field of Borel equivalence relations wasframed by the landmark theorem of Harrington,Kechris, and Louveau.
7
Recall that E0 is the equivalence relation ofeventual agreement on infinite binary sequences.
Theorem 2.3 (Harrington, Kechris, Louveau,1990) Let E be a Borel equivalence relationon a standard Borel space. Then exactly oneof:
1. E !B id(R); or
2. E0 !B E.
This raised the fledgling hope that we might beable to provide a kind of structure theorem forthe Borel equivalence relations under !B, butbefore recounting this part of the tale I wish todescribe the analogies which exist in the theoryof L(R) cardinality.
8
3 Cardinality in L(R)
Definition L(R) is the smallest model of ZFcontaining the reals and the ordinals.
Although this quick formulation finesses out ofthe need to provide any set theoretical formal-ities, it rather disguises the true nature of thisinner model.
It turns out that L(R) can be defined by sim-ply closing the reals under certain kinds of highly“constructive” operations carried out throughtransfinite length along the ordinals. It shouldpossibly be thought of as the collection of setswhich can be defined “internally” or “primi-tively” from the reals and the ordinals.
In this talk I want to think of it as a classinner model which contains anything one mightthink of as being a necessary consequence of theexistence of the reals.
9
It also turns out that ZFC is incapable of de-ciding even the most basic questions about thetheory and structure of L(R).
On the other hand, if L(R) satisfies AD, orthe “Axiom of Determinancy” then almost allthose ambiguities are resolved.
Following work of work of Martin, Steel, Woodin,and others, we now know that any reasonablylarge “large cardinal assumption” implies L(R) |=AD.
This along with the fact that L(R) |= ADhas many regularity properties displayed by theBorel sets (such as the perfect set theorem forarbitrary sets in standard Borel spaces, all setsof reals Lebesgue measurable) has convinced manyset theorists, though not all, that this is the rightassumption under which to explore its structure.
10
I am not going to ask the audience to necessar-ily accept this perspective. I am simply goingto examine the cardinality theory of L(R) as akind of idealization of the theory of Borel cardi-nality.
From now on in this part I will as-sume
L(R) |= AD.
Definition For A and B in L(R), we say thatthe L(R) cardinality of A is less than or equalto the L(R) cardinality of B, written
|A|L(R) ! |B|L(R),
if there is an injection in L(R) from A to B.Similarly
|A|L(R) < |B|L(R),
if there is an injection in L(R) from A to B butnot from B to A.
11
Since the axiom of choice fails inside L(R),there is no reason to imagine that the L(R) car-dinals will be linearly ordered, and in fact thereare incomparable cardinals inside L(R).
It turns out that the theory of L(R) cardinal-ity simulates and extends the theory of Borelcardinality.
In every significant case, the proof that
E <B F
has also given a proof that
|X/E|L(R) < |Y/F |L(R).
This is partially explained by:
Fact 3.1 For E and F Borel equivalence re-lations one has
E !L(R) F
if and only if
|R/E|L(R) ! |R/F |L(R).12
The two dichotomy theorems for Borel equiv-alence relations allow a kind of extension to thecardinality theory of L(R).
Theorem 3.2 (Woodin) Let A ! L(R). Thenexactly one of the following two things musthappen:
1. |A|L(R) " |!|L(R), some ordinal !; or
2. |R|L(R) " |A|L(R).
Theorem 3.3 (Hjorth) Let A ! L(R). Thenexactly one of the following two things musthappen:
1. |A|L(R) " |P(!)|L(R), some ordinal !; or
2. |P(")/Fin|L(R) " |A|L(R).
Here |P(")/Fin|L(R) = |2N/E0|L(R), thus pro-viding an analogy with Harrington-Kechris-Louveau.
13
4 Further structure
Definition Let E1 be the equivalence relationof eventual agreement on RN. For !x, !y ! (2N)N,set !x(E0)
N!y if at every coordinate xnE0yn.
Theorem 4.1 (Kechris, Louveau) Assume
E "B E1.
Then exactly one of:
1. E "B E0; or
2. E1 "B E.
Theorem 4.2 (Hjorth, Kechris) Assume
E "B (E0)N.
Then exactly one of:
1. E "B E0; or
2. (E0)N "B E.
14
Admittedly these are far more local in nature.
These are the only immediate successors to E0which we have established.
There is an entire spectrum of examples, con-structed by Ilijas Farah using ideas from Banachspace theory, for which it seems natural to sup-pose they must be minimal above E0.
However this remains open, due to problemsin the theory of countable Borel equivalence re-lations which appear unattainable using currenttechniques.
Moreover Alexander Kechris and Alain Lou-veau have shown that there is a sense in whichthere are no more global dichotomy theoremsafter Harrington, Kechris, Louveau.
15
5 Anti-structure
Theorem 5.1 (Louveau, Velickovic)There are continuum many many !B in-
comparable Borel equivalence relations.3
In fact we can embed P(N) into !B:
There is an assignment
S "# ES
of Borel equivalence relations to subsets of N
such that for all S, T $ N we have that T \Sis finite if and only if
ET !B ES.
Thus there is nothing like the kind of struc-ture for Borel cardinality that one finds withthe Wadge degrees.
3This first part may have been proved earlier by Hugh Woodin in unpublished work.16
Theorem 5.2 (Kechris, Louveau) There isno Borel E >B E0 with the property that forall other Borel F we always have one of:
1. F !B E; or
2. E !B F .
Two key facts: First of all, Kechris and Lou-veau showed that E1 is not Borel reducible toany EG arising as a result of a continuous Pol-ish group action4, and secondly Leo Harringtonshowed that the Borel EG’s of this form are un-bounded with respect to Borel reducibility:
Theorem 5.3 (Harrington) There is a col-lection {E! : ! " "} of Borel equivalencerelations such:
1. Each E! arises as a result of the continu-ous Polish group action on a Polish space;
2. For any Borel F there will be some ! withE! not Borel reducible to F .
4A theoreom due to Howard Becker and Alexander Kechris theorem on changing topologies in the dynamical context showsthat there is no basically no di!erence between equivalence relations induced by continuous actions and induced by Borelactions. However it is important that the responsible group be a Polish group – there are certain traces of rigidity for Polishgroups, whereas Borel actions of Borel groups can induce any Borel equivalence relation one cares to name
17
To sketch a proof by contradiction of Kechrisand Louveau’s result, suppose E was a Borelequivalence relation with the property that forall Borel F we have one of:
1. F !B E; or
2. E !B F .
Referring back to Harrington’s theorem, therewill be some ! with E! not Borel reducible toE.
Thus since 1 fails for F = E! we must haveE <B E!
But E1 is not Borel reducible to any Polishgroup action, and hence using the same reason-ing we must have E <B E1.
Which by the Kechris-Louveau dichotomy the-orem yields E !B E0.
18
However this proof prompts the following re-sponse:
Question Let E be a Borel equivalence rela-tion. Must we have one of the following:
1. E !B EG some EG induced by the continu-ous action of a Polish group on a Polish space;or
2. E1 !B E?
In other words, is E1 the only obstruction to“classification” or “reduction” to a Polish groupaction?
At present this is wide open.
The question has, however, been positively an-swered by Slawomir Solecki in many special cases.In particular, his penetrating structure theoremfor Polishable ideals proves it for equivalence re-lations on 2N arising as the coset equivalencerelation of some Borel ideal.
19
6 Dichotomy theorems for classification by countable structures
Definition An equivalence relation E on a Pol-ish space X is classifiable by countable struc-tures if there is a countable language L and aBorel function
f : X ! Mod(L)
such that for all x1, x2 " X
x1Ex2 # f (x1) $= f (x2).
This notion of classifiability has been subjectto close scrutiny, in part since it is so naturalfrom the perspective of a logician.5
It might also provide a template of what wecould hope to achieve with other notions of clas-sifiability, where some kind of structure theo-rems can be proved without appeal to a Har-rington, Kechris, Louveau type dichotomy the-orem.
5In fact a Borel equivalence relation E will be L(R) classifiable by countable structures if and only if |X/E|L(R) ! |HC| –classifiability in this sense amounting to reducible to the hereditarily countable sets
20
Theorem 6.1 (Farah ) There is a family ofcontinuum many Borel equivalence relations,(Er)r!R, such that:
1. each Er is induced by the continuous ac-tion of an abelian Polish group on a Polishspace; and
2. no Er is classifiable by countable struc-tures;
3. for r "= s the equivalence relations are in-comparable with respect to Borel reducibil-ity;
4. if E <B Er, any r, then E is essentiallycountable, and hence classifiable by count-able structures.
This says that there is no single canonical ob-struction to be classifiable by countable in theway we find E0 as a canonical obstruction tosmoothness.
21
Definition Let G be a Polish group acting con-tinuously on a Polish space X . For V an openneighborhood of 1G, U an open set containingx, we let
O(x, U, V ),
the U-V -local orbit, be the set of all x ! [x]Gsuch that there is a finite sequence
(xi)i"k # U
such thatx0 = x, xk = x,
and eachxi+1 ! V · xi.
Definition Let G be a Polish group acting con-tinuously on a Polish space X . The action issaid to be turbulent if:
1. every orbit is dense; and
2. every orbit is meager; and
3. for x ! X , the local orbits of x are all some-where dense.
22
Farah’s theorem tells us we can not find evenfinitely many Borel equivalence relations whichare canonical obstructions for classification bycountable structures.
Theorem 6.2 (Hjorth) Let G be a Polish groupacting continuously on a Polish space X withinduced orbit equivalence relation EX
G . As-
sume EXG is Borel.
Then exactly one of:
1. EXG is classifiable by countable structures;
or
2. G acts turbulently on some Polish space Yand
EYG !B EX
G .
There are in fact cases where one can rule outthe existence of turbulent actions by a group,and thus show all the orbit equivalence relationsinduced by a certain Polish group must be clas-sifiable by countable structures.
23
Theorem 6.3 Let G be a Polish group act-ing continuously on a Polish space X withinduced orbit equivalence relation EX
G . As-
sume EXG is Borel.
Then exactly one of:
1. EXG is smooth; or
2. G acts continuously on a Polish space Ywith all orbits dense and meager and
EYG !B EX
G .
Definition If G is a Polish group acting on aPolish space X , we call X stormy if for everynonempty open V " G and x # X the map
V $ [x]G
g %$ g · x
is not an open map.
In a manner parallel to the theory of turbu-lence stormy provides the obstruction for beingessentially countable.
24
7 The wish list
Question Let E be a Borel equivalence rela-tion.
Must we have one of:
1. E !B EG for some EG arising as the orbitequivalence relation of a Polish group actingcontinuously on a Polish space; or
2. E1 !B E?
More generally, if we could establish that thereis some analysis of when an equivalence relationis Borel reducible to a Polish group action, thenwe could lever the theorems regarding turbu-lence and stormy actions to gain a general un-derstanding of when a Borel equivalence relationadmits classification by countable structures oris essentially countable.
25
Question Let EG arise from the continuousaction of an abelian Polish group on a Polishspace. Let E !B EG be a Borel equivalencerelation with countable classes.
Must we then have E !B E0?
If so, then Farah’s earlier examples would ob-tain continuum many immediate successors toE0 in the !B ordering.
Other work of Farah would obtain Borel equiv-alence relations which are above E0 but have noimmediate successor to E0 below.
26
Is there a kind of generalized dichotomy theo-rem for hyperfiniteness?
Most optimistically:
Question Let E be a countable Borel equiva-lence relation. Must we have either:
1. E !B E0; or
2. there is a free measure preserving action ofF2 on a standard Borel probability space suchthat EF2
!B E?
It is known that no such EF2is Borel reducible
to E0.
27
This seems wildly optimistic at present, andperhaps it would be less rash to ask it only inthe case that E is treeable, but it would in par-ticular have as one of its consequences a positiveanswer to the following:
Question Let G be a countable amenable6 group.Suppose G acts in a Borel manner on a standardBorel space X .
Must we have EG !B E0?
The closest result to this is given by a startlinglyoriginal combinatorial argument due to Su Gaoand Steve Jackson who establish a positive an-swer in the case G is abelian.
There are no known techniques, or even hintsat ideas, which could provide a counterexampleto the above question.
6Amenability can be characterized as the statement that for all F ! G finite, ! > 0, there is some A ! G finite with
|A!g · A|
|A|
all g " F .28
All known proofs that an equivalence relationis not reducible to E0 rely on measure theory,and it follows from Connes, Feldman, Weiss thatany such EG must be Borel reducible to E0 onsome conull set with respect to any Borel prob-ability measure.
In fact:
Question Let E be a countable Borel equiv-alence relation. Are measure theoretic reasonsthe only obstruction to being Borel reducibleto E0?
For instance, if E is countable and not Borelreducible to E0, must it be the case that thereis a Borel probability measure µ such that E|Ais not Borel reducible to E0 on any conull A?
Any counterexample would require the devel-opment of fundamentally new ideas about howto prove some equivalence relations are not !BE0.
29
