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Invariant descriptive set theory

Invariant descriptive set theory: an introduction

Invariant descriptive set theory is the study of the complexity ofequivalence relations on standard Borel spaces.

Motivation. In mathematics one often looks for complete invariants toassign to some notion of equivalence:

I Vector spaces on some fixed field are isomorphic iff they have thesame dimension

I Compact orientable surfaces are homeomorphic iff they have thesame genus

I Complex square matrices are similar iff they have the same Jordannormal form

I (Ornstein) Bernoulli shifts are isomorphic if they have the sameentropy

I . . .

I (Baer) To each countable torsion free Abelian group G of rank 1 itis possible to assign a sequence hG of natural numbers s.t. G ∼ G ′

iff hG , hG ′ are eventually equal

I Commutative AF -algebras are isomorphic iff their dimension groupsare isomorphic

I . . .

In order for these assignment to be of some use, they should besufficiently effective.

The right notion of effectiveness turns out to be that of a Borel function,provided the collections of objects to be classified form a standard Borelspace — which is often the case.

I . . .

The right notion of effectiveness turns out to be that of a Borel function,

provided the collections of objects to be classified form a standard Borelspace — which is often the case.

I . . .

The right notion of effectiveness turns out to be that of a Borel function,provided the collections of objects to be classified form a standard Borelspace —

which is often the case.

I . . .

Borel reducibility

DefinitionLet E ,F be equivalence relations on standard Borel spaces X ,Y , resp.Then E is Borel reducible to F , denoted

E ≤B F ,

iff there is f : X → Y Borel s.t.

∀x , x ′ ∈ X (xEx ′ ⇔ f (x)Ff (x ′))

When E ≤B F , any classification of objects in Y up to F -equivalence canbe translated to a classification of objects of X up to E -equivalence, byapplying function f . In this sense, the complexity of E is less than orequal to the complexity of F .

If E ≤B F ≤B E , one writes E ∼B F : the equivalence relations E ,F areBorel bireducible.

Borel reducibility

DefinitionLet E ,F be equivalence relations on standard Borel spaces X ,Y , resp.

Then E is Borel reducible to F , denoted

E ≤B F ,

∀x , x ′ ∈ X (xEx ′ ⇔ f (x)Ff (x ′))

Borel reducibility

E ≤B F ,

∀x , x ′ ∈ X (xEx ′ ⇔ f (x)Ff (x ′))

Borel reducibility

E ≤B F ,

∀x , x ′ ∈ X (xEx ′ ⇔ f (x)Ff (x ′))

When E ≤B F , any classification of objects in Y up to F -equivalence canbe translated to a classification of objects of X up to E -equivalence, byapplying function f .

In this sense, the complexity of E is less than orequal to the complexity of F .

Borel reducibility

E ≤B F ,

∀x , x ′ ∈ X (xEx ′ ⇔ f (x)Ff (x ′))

Borel reducibility

E ≤B F ,

∀x , x ′ ∈ X (xEx ′ ⇔ f (x)Ff (x ′))

Smooth equivalence relations

Among the simplest equivalence relations are those whose classes can becharacterised by a single real number:

DefinitionAn equivalence relation E on a standard Borel X is smooth if E ≤B=,i.e., there exists f : X → R Borel s.t.

∀x , x ′ ∈ X (xEx ′ ⇔ f (x) = f (x ′))

FactGiven two uncountable standard Borel spaces Y ,Z there is always aBorel isomorphism g : Y → Z .

∀x , x ′ ∈ X (xEx ′ ⇔ f (x) = f (x ′))

DefinitionAn equivalence relation E on a standard Borel X is smooth if E ≤B=,

i.e., there exists f : X → R Borel s.t.

∀x , x ′ ∈ X (xEx ′ ⇔ f (x) = f (x ′))

So an equivalence relation is smooth if Borel reduces to equality on someuncountable standard Borel spaces

Using this, the following equivalence relations are smooth:

I Homeomorphism on compact orientable surfaces

I Similarity on complex square matrices

I Isomorphism on Bernoulli shifts

I (Gromov) Isometry on compact metric spaces

I . . .

So an equivalence relation is smooth if Borel reduces to equality on someuncountable standard Borel spacesUsing this, the following equivalence relations are smooth:

I Homeomorphism on compact orientable surfaces

I Similarity on complex square matrices

I Isomorphism on Bernoulli shifts

I (Gromov) Isometry on compact metric spaces

I . . .

Non-smooth equivalence relations

Not all equivalence relations are smooth:

Examples

I Vitali equivalence. On R let xEV y ⇔ x − y ∈ Q.

I Eventual equality on NN. Let xE0y ⇔ ∀∞n x(n) = y(n).

I Isomorphism 'TFA1 for torsion free Abelian groups of rank 1.

It turns out that =R<B EV ∼B E0 ∼B'TFA1 .

Examples

I Vitali equivalence.

On R let xEV y ⇔ x − y ∈ Q.

Examples

I Eventual equality on NN.

Let xE0y ⇔ ∀∞n x(n) = y(n).

Examples

Spaces of countable structures

LetL = {Ri , fj , ck | i ∈ I , j ∈ J, k ∈ K}

be a countable first-order language, where ni ,mj are the arities of Ri , fj ,resp. A countable structure A — say with universe N — is defined byfixing interpretations

I RAi ⊆ Nni , i.e. RAi ∈ 2Nni , for each i ∈ I

I f Aj : Nmj → N, i.e. f Aj ∈ NNmj, for each j ∈ J

I cAk ∈ N, for each k ∈ K

SoA ∈ (

∏i∈I

2Nni)× (

∏j∈J

NNmj)× NK

andXL = (

∏i∈I

2Nni)× (

∏j∈J

NNmj)× NK

is the Polish space of countable L-structures (with universe N)

be a countable first-order language, where ni ,mj are the arities of Ri , fj ,resp.

A countable structure A — say with universe N — is defined byfixing interpretations

SoA ∈ (

∏i∈I

2Nni)× (

∏j∈J

NNmj)× NK

andXL = (

∏i∈I

2Nni)× (

∏j∈J

NNmj)× NK

SoA ∈ (

∏i∈I

2Nni)× (

∏j∈J

NNmj)× NK

andXL = (

∏i∈I

2Nni)× (

∏j∈J

NNmj)× NK

I RAi ⊆ Nni ,

i.e. RAi ∈ 2Nni , for each i ∈ I

SoA ∈ (

∏i∈I

2Nni)× (

∏j∈J

NNmj)× NK

andXL = (

∏i∈I

2Nni)× (

∏j∈J

NNmj)× NK

I f Aj : Nmj → N,

i.e. f Aj ∈ NNmj, for each j ∈ J

SoA ∈ (

∏i∈I

2Nni)× (

∏j∈J

NNmj)× NK

andXL = (

∏i∈I

2Nni)× (

∏j∈J

NNmj)× NK

SoA ∈ (

∏i∈I

2Nni)× (

∏j∈J

NNmj)× NK

andXL = (

∏i∈I

2Nni)× (

∏j∈J

NNmj)× NK

SoA ∈ (

∏i∈I

2Nni)× (

∏j∈J

NNmj)× NK

andXL = (

∏i∈I

2Nni)× (

∏j∈J

NNmj)× NK

SoA ∈ (

∏i∈I

2Nni)× (

∏j∈J

NNmj)× NK

andXL = (

∏i∈I

2Nni)× (

∏j∈J

NNmj)× NK

The logic action

To simplify notation, from now on I will assume that L is relational, i.e.,

XL =∏i∈I

Elements of XL will be usually denoted by letters like x , y , . . ..

Let S∞ = Sym(N) be the group of permutations of N. This is a Gδsubset of NN: for every g ∈ NN, one has g ∈ S∞ iff

1. g is surjective: ∀m ∃n g(n) = m

2. g is injective: ∀n,m (n 6= m⇒ g(n) 6= g(m))

So S∞ is a Polish space. Moreover, the operations of composition(g , h) 7→ gh and inversion g 7→ g−1 are continuous in S∞, so it is aPolish group.

The logic action

XL =∏i∈I

Elements of XL will be usually denoted by letters like x , y , . . ..Let S∞ = Sym(N) be the group of permutations of N.

This is a Gδsubset of NN: for every g ∈ NN, one has g ∈ S∞ iff

The logic action

XL =∏i∈I

Elements of XL will be usually denoted by letters like x , y , . . ..Let S∞ = Sym(N) be the group of permutations of N. This is a Gδsubset of NN:

for every g ∈ NN, one has g ∈ S∞ iff

The logic action

XL =∏i∈I

Elements of XL will be usually denoted by letters like x , y , . . ..Let S∞ = Sym(N) be the group of permutations of N. This is a Gδsubset of NN: for every g ∈ NN, one has g ∈ S∞ iff

The logic action

XL =∏i∈I

The logic action

XL =∏i∈I

So S∞ is a Polish space.

Moreover, the operations of composition(g , h) 7→ gh and inversion g 7→ g−1 are continuous in S∞, so it is aPolish group.

The logic action

XL =∏i∈I

So S∞ is a Polish space. Moreover, the operations of composition(g , h) 7→ gh and inversion g 7→ g−1 are continuous in S∞,

so it is aPolish group.

The logic action

XL =∏i∈I

The logic action

The action of S∞ on N induces an action of S∞ on XL, the logic action:

gx = y ⇔ ∀i yi (h1, . . . , hn) = xi (g−1(h1), . . . , g−1(hn))

i.e., gx = y iff g : N→ N is an isomorphism between the L-structuresx , y . The orbit equivalence relation is isomorphism:x ' y ⇔ ∃g ∈ S∞ gx = y .

DefinitionA subset A of a standard Borel space X is analytic if it is the Borel imageof a Borel subset of a standard Borel space:

A ∈ Σ11(X )⇔ ∃f : Y → X Borel ∃B ∈ B(Y ) f (B) = A

So the isomorphism relation ' on XL is analytic, as a subset of X 2L (but

in general not Borel).

The logic action

i.e., gx = y iff g : N→ N is an isomorphism between the L-structuresx , y .

The orbit equivalence relation is isomorphism:x ' y ⇔ ∃g ∈ S∞ gx = y .

The logic action

Borel sets in XL

However, the following holds:

Theorem (Miller)Let G be a Polish group, X a standard Borel space, (g , x) 7→ gx a Borelaction of G on X .

Then every orbit {gx}g∈G is Borel.

DefinitionLω1ω is the extension of L obtained by allowing countable disjunctionsand conjunctions

∨nϕn,∧nϕn

where each ϕn has free variables between v0, . . . , vk−1 for some kindependent of n (so each formula has finitely many free variables).

Borel sets in XL

Theorem (Miller)Let G be a Polish group, X a standard Borel space, (g , x) 7→ gx a Borelaction of G on X . Then every orbit {gx}g∈G is Borel.

∨nϕn,∧nϕn

Borel sets in XL

∨nϕn,∧nϕn

Borel sets in XL

∨nϕn,∧nϕn

Borel sets in XL

PropositionLet ϕ(v0, . . . , vk−1) be a formula of Lω1ω.

Then the set Aϕ,k ⊆ XL × Nk

defined by

(x , a0, . . . , ak−1) ∈ Ak ⇔ x |= ϕ(a0, . . . , ak−1)

is Borel.

Proof.By induction on the construction of ϕ.

Borel sets in XL

PropositionLet ϕ(v0, . . . , vk−1) be a formula of Lω1ω. Then the set Aϕ,k ⊆ XL × Nk

defined by

(x , a0, . . . , ak−1) ∈ Ak ⇔ x |= ϕ(a0, . . . , ak−1)

is Borel.

Borel sets in XL

defined by

(x , a0, . . . , ak−1) ∈ Ak ⇔ x |= ϕ(a0, . . . , ak−1)

is Borel.

Borel sets in XL

defined by

(x , a0, . . . , ak−1) ∈ Ak ⇔ x |= ϕ(a0, . . . , ak−1)

is Borel.

Borel sets in XL

In particular, if σ is a sentence of Lω1ω, i.e., k = 0, thenAσ = Aσ0 = {x | x |= σ} is Borel

and invariant under '.

The converse is true as well:

Theorem (Lopez-Escobar)The invariant Borel subsets of XL are exactly those of the form Aσ, for σa sentence of Lω1ω.

Corollary (Scott)For every countable L-structure A there is a sentence σA of Lω1ω suchthat for any countable L-structure B, one has

B ' A iff B |= σA

(such a sentence is called a Scott sentence of A)

Borel sets in XL

In particular, if σ is a sentence of Lω1ω, i.e., k = 0, thenAσ = Aσ0 = {x | x |= σ} is Borel and invariant under '.

B ' A iff B |= σA

Borel sets in XL

B ' A iff B |= σA

Borel sets in XL

B ' A iff B |= σA

Borel sets in XL

B ' A iff B |= σA

Borel sets in XL

FactA Borel subset of a standard Borel space, with the induced σ-algebra, isitself standard Borel.

In particular, the usual classes of countable structures are Borel andinvariant in XL (where L is a suitable language), so they form standardBorel spaces:

I graphs: L = {R}, where R is a binary relation symbol

I groups: L = {·,−1, 1}, where · is a binary function symbol, −1 is aunary function symbol, 1 is a constant symbol

I fields: L = . . .

I linear orders: L = {<}, where < is a binary relation symbol

I . . .

Borel sets in XL

I fields: L = . . .

I . . .

Borel sets in XL

I fields: L = . . .

I . . .

Borel sets in XL

I fields: L = . . .

I . . .

Borel sets in XL

I fields: L = . . .

I . . .

Borel sets in XL

I fields: L = . . .

I . . .

Classification by countable structures

DefinitionAn equivalence relation E is classifiable by countable structures if there isan invariant Borel class Z of countable structures such that E is Borelreducible to isomorphism on Z :

E ≤B'Z

Examples.

I Isomorphism on commutative AF algebras

I Isometry on ultrametric Polish spaces

I (Darji, Marcone, C; 2005) Homeomorphism on dendrites (compact,connected, locally connected metric spaces not containing simpleclosed curves)

I . . .

E ≤B'Z

Examples.

I . . .

E ≤B'Z

Examples.

I . . .

E ≤B'Z

Examples.

I . . .

E ≤B'Z

Examples.

I . . .

Complete elements

DefinitionIf C is a class of equivalence relations on standard Borel spaces, anequivalence relation E is C-complete if:

1. E ∈ C2. ∀F ∈ C F ≤B E

It turns out that several classes C of equivalence relations have a completemember. The interest is to find examples of C-complete relations, or ofrelations that are not C-complete but not for trivial reasons.

Complete elements

1. E ∈ C2. ∀F ∈ C F ≤B E

Complete elements

1. E ∈ C2. ∀F ∈ C F ≤B E

Complete elements

1. E ∈ C2. ∀F ∈ C F ≤B E

It turns out that several classes C of equivalence relations have a completemember.

The interest is to find examples of C-complete relations, or ofrelations that are not C-complete but not for trivial reasons.

Complete elements

1. E ∈ C2. ∀F ∈ C F ≤B E

S∞-complete equivalence relations

If C is the class of equivalence relations classifiable by countablestructures, a C-complete equivalence relation is also called S∞-complete.

Examples of S∞-equivalence relations

I The isomorphism relation on countable graphs, or on countable trees

I (Gao, C.; 2001) The isomorphism relation on countable Booleanalgebras; the homeomorphism relation on zero-dimensional compactmetric spaces; the conjugacy relation on the group ofhomeomorphism of Cantor space

I (Darji, Marcone, C; 2005) The homeomorphism relation on dendrites

Analytic equivalence relations

The preorder ≤B is defined between all equivalence relations on standardBorel spaces. However, the class Σ1

1 of analytic equivalence relations is,up to date, the biggest class for which some structural properties andsignificant examples could be found.

TheoremThere exists a Σ1

1-complete equivalence relation.

The preorder ≤B is defined between all equivalence relations on standardBorel spaces. However, the class Σ1

1 of analytic equivalence relations is,up to date, the biggest class for which some structural properties andsignificant examples could be found.

TheoremThere exists a Σ1

1-complete equivalence relation.

Examples of Σ11-complete equivalence relations

I (Louveau, Rosendal; 2005) Biembeddability on countable trees

I (Ferenczi, Louveau, Rosendal) Isomorphism on separable Banachspaces

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