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A descriptive view of infinite dimensional group representations Simon Thomas Rutgers University Quae ministratur a capite pulli 26th May 2016 Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016 Finite Dimensional Representations

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A descriptive view of infinite dimensional group representations

Simon Thomas

Rutgers University “Quae ministratur a capite pulli”

26th May 2016

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Finite Dimensional Representations

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Finite Dimensional Representations

Definition

If G is a finite group, then a linear representation of G is a homomorphism ϕ : G → GLn(C) for some n ≥ 1.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Finite Dimensional Representations

Definition

If G is a finite group, then a linear representation of G is a homomorphism ϕ : G → GLn(C) for some n ≥ 1.

Definition

Two representations ϕ : G → GLn(C) and ψ : G → GLm(C) are equivalent if n = m and there exists A ∈ GLn(C) such that ψ(g) = A ϕ(g) A−1 for all g ∈ G.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Finite Dimensional Irreducible Representations

Definition

The representation ϕ : G → GLn(C) is irreducible if there are no nontrivial proper G-invariant subspaces 0 < W < Cn.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Finite Dimensional Irreducible Representations

Definition

The representation ϕ : G → GLn(C) is irreducible if there are no nontrivial proper G-invariant subspaces 0 < W < Cn.

Theorem (Frobenius & Burnside 1904)

If G is a finite group, then the number of irreducible representations

  • f G (up to equivalence) is equal to the number of conjugacy classes.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Arbitary Finite Dimensional Representations

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Arbitary Finite Dimensional Representations

Theorem (Maschke & Schur 1905)

If G is a finite group, then every representation of G is uniquely expressible as a direct sum of irreducible representations.

Definition

If ϕ : G → GLn(C) and ψ : G → GLm(C) are representations, then the direct sum (ϕ ⊕ ψ) : G → GLn+m(C) is defined by g → ϕ(g) ψ(g)

  • Simon Thomas (Rutgers)

Lezione Lagrangiana 26th May 2016

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Finite Dimensional Unitary Representations

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Finite Dimensional Unitary Representations

Definition

The unitary group Un(C) is the subgroup of GLn(C) which preserves the inner product on Cn defined by u, v = u1¯ v1 + · · · + un¯ vn;

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Finite Dimensional Unitary Representations

Definition

The unitary group Un(C) is the subgroup of GLn(C) which preserves the inner product on Cn defined by u, v = u1¯ v1 + · · · + un¯ vn;

  • r equivalently which preserve the norm on Cn defined by

|| u || =

  • u, u =
  • |u1|2 + · · · + |un|2

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Finite Dimensional Unitary Representations

Definition

The unitary group Un(C) is the subgroup of GLn(C) which preserves the inner product on Cn defined by u, v = u1¯ v1 + · · · + un¯ vn;

  • r equivalently which preserve the norm on Cn defined by

|| u || =

  • u, u =
  • |u1|2 + · · · + |un|2

If G is a finite group, then a unitary representation of G is a homomorphism ϕ : G → Un(C) for some n ≥ 1.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Finite Dimensional Unitary Representations

Observation

If ϕ : G → Un(C) is a unitary representation and W Cn is G-invariant, then so is the orthogonal complement W ⊥ = { v ∈ Cn | v, w = 0 for all w ∈ W }.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Finite Dimensional Unitary Representations

Observation

If ϕ : G → Un(C) is a unitary representation and W Cn is G-invariant, then so is the orthogonal complement W ⊥ = { v ∈ Cn | v, w = 0 for all w ∈ W }.

Theorem (Loewy & Moore 1898)

If G is a finite group, then every representation of G is equivalent to a unitary representation.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Finite Dimensional Unitary Representations

Observation

If ϕ : G → Un(C) is a unitary representation and W Cn is G-invariant, then so is the orthogonal complement W ⊥ = { v ∈ Cn | v, w = 0 for all w ∈ W }.

Theorem (Loewy & Moore 1898)

If G is a finite group, then every representation of G is equivalent to a unitary representation.

Theorem (Folklore)

If G is a finite group and ϕ, ψ : G → Un(C) are unitary representations, then ϕ, ψ are equivalent iff ϕ, ψ are unitarily equivalent.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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How about representations of infinite groups?

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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How about representations of infinite groups?

Theorem

Alt(N) = Alt(m) has no nontrivial finite dimensional representations.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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How about representations of infinite groups?

Theorem

Alt(N) = Alt(m) has no nontrivial finite dimensional representations.

Proof.

If m ≥ 6 and ϕ : Alt(m) → GLn(C) is a nontrivial representation, then n ≥ m − 1.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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How about representations of infinite groups?

Theorem

Alt(N) = Alt(m) has no nontrivial finite dimensional representations.

Proof.

If m ≥ 6 and ϕ : Alt(m) → GLn(C) is a nontrivial representation, then n ≥ m − 1.

Question

How about infinite dimensional representations of infinite groups?

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Infinite Dimensional Representations

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Infinite Dimensional Representations

The Separable Hilbert Spaces

If X is a countable set, then ℓ2(X) = { a : X → C :

  • |a(x)|2 < ∞ },

equipped with the inner product a, b =

  • a(x)b(x).

and the corresponding norm || a || =

  • |a(x)|2.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Infinite Dimensional Representations

Definition

If G is a countable group, then a unitary representation of G is a homomorphism ϕ : G → U(H), where: H is a separable complex Hilbert space; and U(H) is the corresponding group of unitary transformations.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Infinite Dimensional Representations

Definition

If G is a countable group, then a unitary representation of G is a homomorphism ϕ : G → U(H), where: H is a separable complex Hilbert space; and U(H) is the corresponding group of unitary transformations.

Example

We can define a unitary representation on ℓ2(G) = { a : G → C :

  • |a(x)|2 < ∞ }

by letting (g · a)(x) = a(g−1x).

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Unitary Representations of Countable Groups

Definition

Two representations ϕ : G → U(H) and ψ : G → U(H) are unitarily equivalent if there exists A ∈ U(H) such that ψ(g) = A ϕ(g) A−1 for all g ∈ G.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Unitary Representations of Countable Groups

Definition

Two representations ϕ : G → U(H) and ψ : G → U(H) are unitarily equivalent if there exists A ∈ U(H) such that ψ(g) = A ϕ(g) A−1 for all g ∈ G.

Definition

The unitary representation ϕ : G → U(H) is irreducible if there are no nontrivial proper G-invariant closed subspaces 0 < W < H.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Unitary Representations of Countable Groups

Problem

Can we classify the irreducible unitary representations of G up to unitary equivalence?

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Unitary Representations of Countable Groups

Problem

Can we classify the irreducible unitary representations of G up to unitary equivalence? Can we classify arbitrary unitary representations of G via “suitable decompositions” into irreducible representations?

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The Unitary Representations of Z

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The Unitary Representations of Z

According to Shelah:

“ I have always felt that examples usually just confuse you, having always specific properties that are traps as they do not hold in general.”

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The Unitary Representations of Z

According to Shelah:

“ I have always felt that examples usually just confuse you, having always specific properties that are traps as they do not hold in general.”

Theorem (Folklore)

The irreducible unitary representations of Z are ϕz : Z → U1(C) = T = { c ∈ C : |c| = 1 } where z ∈ T and ϕz(k) = zk.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The Unitary Representations of Z

According to Shelah:

“ I have always felt that examples usually just confuse you, having always specific properties that are traps as they do not hold in general.”

Theorem (Folklore)

The irreducible unitary representations of Z are ϕz : Z → U1(C) = T = { c ∈ C : |c| = 1 } where z ∈ T and ϕz(k) = zk.

Observation

The unitary representation Z ℓ2(Z) has no Z-invariant 1-dimensional subspaces.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The Unitary Representations of Z

The “multiplicity-free” unitary representations of Z can be parameterized by the probability measures µ on T so that the following are equivalent:

(ii) the measures µ, ν have the same null sets; (i) the corresponding unitary representations ϕµ, ϕν are unitarily equivalent.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The Unitary Representations of Z

The “multiplicity-free” unitary representations of Z can be parameterized by the probability measures µ on T so that the following are equivalent:

(ii) the measures µ, ν have the same null sets; (i) the corresponding unitary representations ϕµ, ϕν are unitarily equivalent.

Mackey (1955): it is not clear that these measure equivalence classes can be parameterized by the points of a Polish space.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The Unitary Representations of Z

The “multiplicity-free” unitary representations of Z can be parameterized by the probability measures µ on T so that the following are equivalent:

(ii) the measures µ, ν have the same null sets; (i) the corresponding unitary representations ϕµ, ϕν are unitarily equivalent.

Mackey (1955): it is not clear that these measure equivalence classes can be parameterized by the points of a Polish space.

Definition

A Polish space is a separable completely metrizable topological space. E.g. R, C, 2N, NN,...

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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What is a parameterization?

Notation

If E is an equivalence relation on the set X, then X/E denotes the set of E-equivalence classes.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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What is a parameterization?

Notation

If E is an equivalence relation on the set X, then X/E denotes the set of E-equivalence classes.

A First Approximation

If E is an equivalence relation on the set X, then a parameterization

  • f X/E is an explicit map f : X → Z to a Polish space Z such that

f(x) = f(y) ⇐ ⇒ x E y.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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What is an explicit map?

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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What is an explicit map?

Question

Which functions f : R → R are explicit?

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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What is an explicit map?

Question

Which functions f : R → R are explicit?

Church’s Thesis for Real Mathematics

EXPLICIT = BOREL

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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What is an explicit map?

Question

Which functions f : R → R are explicit?

Church’s Thesis for Real Mathematics

EXPLICIT = BOREL

Definition

A function f : R → R is Borel if graph(f) is a Borel subset of R × R.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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What is an explicit map?

Question

Which functions f : R → R are explicit?

Church’s Thesis for Real Mathematics

EXPLICIT = BOREL

Definition

A function f : R → R is Borel if graph(f) is a Borel subset of R × R. Equivalently, if f −1(A) is Borel for each Borel subset A ⊆ R.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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What is a parameterization?

Definition (Mackey)

An equivalence relation E on a Polish space X is parameterizable

  • r smooth if there exists a Borel map f : X → Z to a Polish space Z

such that f(x) = f(y) ⇐ ⇒ x E y.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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What is a parameterization?

Definition (Mackey)

An equivalence relation E on a Polish space X is parameterizable

  • r smooth if there exists a Borel map f : X → Z to a Polish space Z

such that f(x) = f(y) ⇐ ⇒ x E y.

Theorem (Jordan 1870)

The similarity relation on the space Matn(C) of complex n × n matrices is smooth.

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What is a parameterization?

Definition (Mackey)

An equivalence relation E on a Polish space X is parameterizable

  • r smooth if there exists a Borel map f : X → Z to a Polish space Z

such that f(x) = f(y) ⇐ ⇒ x E y.

Theorem (Jordan 1870)

The similarity relation on the space Matn(C) of complex n × n matrices is smooth.

Theorem (Freedman 1966)

If X is an uncountable Polish space, then the measure equivalence relation on the space M(X) of probability measures on X is not smooth.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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A “simple” non-smooth equivalence relation

Definition

E0 is the equivalence relation on 2N defined by: x E0 y ⇐ ⇒ xn = yn for all but finitely many n.

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A “simple” non-smooth equivalence relation

Definition

E0 is the equivalence relation on 2N defined by: x E0 y ⇐ ⇒ xn = yn for all but finitely many n.

Notation

Let µ be the uniform product probability measure on 2N.

Theorem (Kolmogorov Zero-One Law 1933)

If f : 2N → [ 0, 1 ] is a Borel map which is constant on E0-classes, then f is constant on a µ-measure 1 subset.

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Borel reductions

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Borel reductions

Definition (Friedman-Stanley 1989)

Let E, F be equivalence relations on the Polish spaces X, Y.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Borel reductions

Definition (Friedman-Stanley 1989)

Let E, F be equivalence relations on the Polish spaces X, Y. E ≤B F if there exists a Borel map ϕ : X → Y such that x E y ⇐ ⇒ ϕ(x) F ϕ(y). In this case, f is called a Borel reduction from E to F.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Borel reductions

Definition (Friedman-Stanley 1989)

Let E, F be equivalence relations on the Polish spaces X, Y. E ≤B F if there exists a Borel map ϕ : X → Y such that x E y ⇐ ⇒ ϕ(x) F ϕ(y). In this case, f is called a Borel reduction from E to F. E ∼B F if both E ≤B F and F ≤B E.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Borel reductions

Definition (Friedman-Stanley 1989)

Let E, F be equivalence relations on the Polish spaces X, Y. E ≤B F if there exists a Borel map ϕ : X → Y such that x E y ⇐ ⇒ ϕ(x) F ϕ(y). In this case, f is called a Borel reduction from E to F. E ∼B F if both E ≤B F and F ≤B E. E <B F if both E ≤B F and E ≁B F.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Borel reductions

Definition (Friedman-Stanley 1989)

Let E, F be equivalence relations on the Polish spaces X, Y. E ≤B F if there exists a Borel map ϕ : X → Y such that x E y ⇐ ⇒ ϕ(x) F ϕ(y). In this case, f is called a Borel reduction from E to F. E ∼B F if both E ≤B F and F ≤B E. E <B F if both E ≤B F and E ≁B F.

Remark

In particular, an equivalence relation E is smooth iff E is Borel reducible to the identity relation ∆Z on some Polish space Z.

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The Glimm-Effos Dichotomy 1961/65

Theorem (Harrington-Kechris-Louveau 1990)

If E is a Borel equivalence relation on the Polish space X, then exactly one of the following holds: (i) E is smooth; or (ii) E0 ≤B E.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The Glimm-Effos Dichotomy 1961/65

Theorem (Harrington-Kechris-Louveau 1990)

If E is a Borel equivalence relation on the Polish space X, then exactly one of the following holds: (i) E is smooth; or (ii) E0 ≤B E.

Definition

An equivalence relation E on a Polish space X is Borel if E is a Borel subset of X × X.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The unitary equivalence relation

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The unitary equivalence relation

Let G be a countably infinite group.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The unitary equivalence relation

Let G be a countably infinite group. Let H be a fixed infinite dimensional separable Hilbert space and let U(H) be the corresponding unitary group.

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The unitary equivalence relation

Let G be a countably infinite group. Let H be a fixed infinite dimensional separable Hilbert space and let U(H) be the corresponding unitary group. Let Irr∞(G) be the Polish space of irreducible representations ϕ : G → U(H).

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The unitary equivalence relation

Let G be a countably infinite group. Let H be a fixed infinite dimensional separable Hilbert space and let U(H) be the corresponding unitary group. Let Irr∞(G) be the Polish space of irreducible representations ϕ : G → U(H). Let ≈G be the unitary equivalence relation defined on Irr∞(G) by ϕ ≈G ψ ⇐ ⇒ ( ∃A ∈ U(H) ) ( ∀g ∈ G ) Aϕ(g)A−1 = ψ(g).

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The unitary equivalence relation

Let G be a countably infinite group. Let H be a fixed infinite dimensional separable Hilbert space and let U(H) be the corresponding unitary group. Let Irr∞(G) be the Polish space of irreducible representations ϕ : G → U(H). Let ≈G be the unitary equivalence relation defined on Irr∞(G) by ϕ ≈G ψ ⇐ ⇒ ( ∃A ∈ U(H) ) ( ∀g ∈ G ) Aϕ(g)A−1 = ψ(g).

Question

Is ≈G a smooth equivalence relation?

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The Glimm-Thoma Theorem

Theorem (Glimm & Thoma 1964)

If G is a countable group, then the following conditions are equivalent:

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The Glimm-Thoma Theorem

Theorem (Glimm & Thoma 1964)

If G is a countable group, then the following conditions are equivalent: (i) G is not abelian-by-finite.

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The Glimm-Thoma Theorem

Theorem (Glimm & Thoma 1964)

If G is a countable group, then the following conditions are equivalent: (i) G is not abelian-by-finite. (ii) G has an infinite dimensional irreducible representation.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The Glimm-Thoma Theorem

Theorem (Glimm & Thoma 1964)

If G is a countable group, then the following conditions are equivalent: (i) G is not abelian-by-finite. (ii) G has an infinite dimensional irreducible representation. (iii) The unitary equivalence relation ≈G on the space Irr∞(G)

  • f infinite dimensional irreducible unitary representations
  • f G is not smooth.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The Glimm-Thoma Theorem

Theorem (Glimm & Thoma 1964)

If G is a countable group, then the following conditions are equivalent: (i) G is not abelian-by-finite. (ii) G has an infinite dimensional irreducible representation. (iii) The unitary equivalence relation ≈G on the space Irr∞(G)

  • f infinite dimensional irreducible unitary representations
  • f G is not smooth.

Question

Does this mean that we should abandon all hope of finding a “satisfactory classification” for the irreducible unitary representations of these groups?

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When it’s bad, it’s worse ...

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When it’s bad, it’s worse ...

Theorem (Hjorth 1997)

If the countable group G is not abelian-by-finite, then the action

  • f U(H) on Irr∞(G) is turbulent.

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When it’s bad, it’s worse ...

Theorem (Hjorth 1997)

If the countable group G is not abelian-by-finite, then the action

  • f U(H) on Irr∞(G) is turbulent.

Remarks

This is a much more serious obstruction to the existence

  • f a “satisfactory classification” of the irreducible unitary

representations of G.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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When it’s bad, it’s worse ...

Theorem (Hjorth 1997)

If the countable group G is not abelian-by-finite, then the action

  • f U(H) on Irr∞(G) is turbulent.

Remarks

This is a much more serious obstruction to the existence

  • f a “satisfactory classification” of the irreducible unitary

representations of G. But hopefully this is not the end of the story ...

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An aside: Classifying homeomorphisms of [0, 1]

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An aside: Classifying homeomorphisms of [0, 1]

Definition

Hom+([0, 1]) is the group of homeomorphisms ϕ : [0, 1] → [0, 1] satisfying ϕ(0) = 0 and ϕ(1) = 1.

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An aside: Classifying homeomorphisms of [0, 1]

Definition

Hom+([0, 1]) is the group of homeomorphisms ϕ : [0, 1] → [0, 1] satisfying ϕ(0) = 0 and ϕ(1) = 1. Two maps ϕ, ψ ∈ Hom+([0, 1]) are conjugate if there exists θ ∈ Hom+([0, 1]) such that ψ = θ ◦ ϕ ◦ θ−1. [0, 1]

θ

− − − − → [0, 1]

ϕ

 

 ψ [0, 1]

θ

− − − − → [0, 1]

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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An aside: Classifying homeomorphisms of [0, 1]

Definition

Hom+([0, 1]) is the group of homeomorphisms ϕ : [0, 1] → [0, 1] satisfying ϕ(0) = 0 and ϕ(1) = 1. Two maps ϕ, ψ ∈ Hom+([0, 1]) are conjugate if there exists θ ∈ Hom+([0, 1]) such that ψ = θ ◦ ϕ ◦ θ−1. [0, 1]

θ

− − − − → [0, 1]

ϕ

 

 ψ [0, 1]

θ

− − − − → [0, 1]

Problem

Classify the elements of Hom+([0, 1]) up to conjugacy by “discrete invariants”.

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The Bump Structure

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The Bump Structure

Definition

A bump of ϕ ∈ Hom+([0, 1]) is a maximal open interval I ⊂ [0, 1] such that one of the following conditions hold: (a) ϕ(x) > x for all x ∈ I. (b) ϕ(x) < x for all x ∈ I. (c) ϕ(x) = x for all x ∈ I.

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The Bump Structure

Definition

A bump of ϕ ∈ Hom+([0, 1]) is a maximal open interval I ⊂ [0, 1] such that one of the following conditions hold: (a) ϕ(x) > x for all x ∈ I. (b) ϕ(x) < x for all x ∈ I. (c) ϕ(x) = x for all x ∈ I.

  • Simon Thomas (Rutgers)

Lezione Lagrangiana 26th May 2016

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The Bump Structure

Definition

A bump of ϕ ∈ Hom+([0, 1]) is a maximal open interval I ⊂ [0, 1] such that one of the following conditions hold: (a) ϕ(x) > x for all x ∈ I. (b) ϕ(x) < x for all x ∈ I. (c) ϕ(x) = x for all x ∈ I.

  • I1

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The Bump Structure

Definition

A bump of ϕ ∈ Hom+([0, 1]) is a maximal open interval I ⊂ [0, 1] such that one of the following conditions hold: (a) ϕ(x) > x for all x ∈ I. (b) ϕ(x) < x for all x ∈ I. (c) ϕ(x) = x for all x ∈ I.

  • I1

I2

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The Bump Structure

Definition

A bump of ϕ ∈ Hom+([0, 1]) is a maximal open interval I ⊂ [0, 1] such that one of the following conditions hold: (a) ϕ(x) > x for all x ∈ I. (b) ϕ(x) < x for all x ∈ I. (c) ϕ(x) = x for all x ∈ I.

  • I1

I2 I3

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The Bump Structure

Definition

A bump of ϕ ∈ Hom+([0, 1]) is a maximal open interval I ⊂ [0, 1] such that one of the following conditions hold: (a) ϕ(x) > x for all x ∈ I. (b) ϕ(x) < x for all x ∈ I. (c) ϕ(x) = x for all x ∈ I.

  • I1

I2 I3 I4

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The Bump Structure

Definition

A bump of ϕ ∈ Hom+([0, 1]) is a maximal open interval I ⊂ [0, 1] such that one of the following conditions hold: (a) ϕ(x) > x for all x ∈ I. (b) ϕ(x) < x for all x ∈ I. (c) ϕ(x) = x for all x ∈ I.

  • I1

I2 I3 I4 I5

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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The Bump Structure

Definition

A bump of ϕ ∈ Hom+([0, 1]) is a maximal open interval I ⊂ [0, 1] such that one of the following conditions hold: (a) ϕ(x) > x for all x ∈ I. (b) ϕ(x) < x for all x ∈ I. (c) ϕ(x) = x for all x ∈ I.

  • I1

I2 I3 I4 I5

Theorem (Folklore)

Two maps ϕ, ψ ∈ Hom+([0, 1]) are conjugate iff the corresponding colored linear orders are isomorphic.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Classifying homeomorphisms of [0, 1]2

Definition

Hom+([0, 1]2) is the group of homeomorphisms ϕ : [0, 1]2 → [0, 1]2 satisfying ϕ(v) = v for each v ∈ {(0, 0), (0, 1), (1, 0), (1, 1)}.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Classifying homeomorphisms of [0, 1]2

Definition

Hom+([0, 1]2) is the group of homeomorphisms ϕ : [0, 1]2 → [0, 1]2 satisfying ϕ(v) = v for each v ∈ {(0, 0), (0, 1), (1, 0), (1, 1)}.

Problem

Classify the elements of Hom+([0, 1]2) up to conjugacy by “discrete invariants”.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Classifying homeomorphisms of [0, 1]2

Definition

Hom+([0, 1]2) is the group of homeomorphisms ϕ : [0, 1]2 → [0, 1]2 satisfying ϕ(v) = v for each v ∈ {(0, 0), (0, 1), (1, 0), (1, 1)}.

Problem

Classify the elements of Hom+([0, 1]2) up to conjugacy by “discrete invariants”. Perhaps there exists a “blister structure”?

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Classifying homeomorphisms of [0, 1]2

Definition

Hom+([0, 1]2) is the group of homeomorphisms ϕ : [0, 1]2 → [0, 1]2 satisfying ϕ(v) = v for each v ∈ {(0, 0), (0, 1), (1, 0), (1, 1)}.

Problem

Classify the elements of Hom+([0, 1]2) up to conjugacy by “discrete invariants”. Perhaps there exists a “blister structure”?

Theorem (Hjorth 1997)

The conjugacy relation on Hom+([0, 1]2) does not admit classification by “discrete invariants” ...

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Classifying homeomorphisms of [0, 1]2

Definition

Hom+([0, 1]2) is the group of homeomorphisms ϕ : [0, 1]2 → [0, 1]2 satisfying ϕ(v) = v for each v ∈ {(0, 0), (0, 1), (1, 0), (1, 1)}.

Problem

Classify the elements of Hom+([0, 1]2) up to conjugacy by “discrete invariants”. Perhaps there exists a “blister structure”?

Theorem (Hjorth 1997)

The conjugacy relation on Hom+([0, 1]2) does not admit classification by “discrete invariants” ... because the conjugacy relation is turbulent.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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When it’s bad, it’s worse ...

Theorem (Hjorth 1997)

If the countable group G is not abelian-by-finite , then the action of U(H) on Irr∞(G) is turbulent.

Remarks

This rules out the existence of a “satisfactory classification”

  • f the irreducible unitary representations of G.

But hopefully this is not the end of the story ...

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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When it’s bad, it’s worse ...

Theorem (Hjorth 1997)

If the countable group G is not abelian-by-finite , then the action of U(H) on Irr∞(G) is turbulent.

Open Question (Thomas 2011)

Do there exist countable groups G, H such that (i) G, H are not abelian-by-finite; and (ii) ≈G, ≈H are not Borel bireducible?

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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When it’s bad, it’s worse ...

Theorem (Hjorth 1997)

If the countable group G is not abelian-by-finite , then the action of U(H) on Irr∞(G) is turbulent.

Open Question (Thomas 2011 & Effros 2007)

Do there exist countable groups G, H such that (i) G, H are not abelian-by-finite; and (ii) ≈G, ≈H are not Borel bireducible?

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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When it’s bad, it’s worse ...

Theorem (Hjorth 1997)

If the countable group G is not abelian-by-finite , then the action of U(H) on Irr∞(G) is turbulent.

Open Question (Thomas 2011 & Effros 2007 & Dixmier 1967)

Do there exist countable groups G, H such that (i) G, H are not abelian-by-finite; and (ii) ≈G, ≈H are not Borel bireducible?

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Dixmier’s Question

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Dixmier’s Question

Question (Dixmier 1967)

Do there exist countable groups G, H such that (i) G, H are not abelian-by-finite; and (ii) the unitary duals Irr∞(G)/ ≈G and Irr∞(H)/ ≈H are not Borel isomorphic?

Definition

If E, F are equivalence relations on the Polish spaces X, Y, then X/E, Y/F are Borel isomorphic if there exist Borel maps ϕ : X → Y and ψ : Y → X which induce mutually inverse bijections ˆ ϕ, ˆ ψ between X/E and Y/F.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Borel bireducibility vs Borel isomorphism

Theorem (Motto Ros 2012)

If E, F are Borel orbit equivalence relations of actions of Polish groups

  • n the Polish spaces X, Y, then the following are equivalent:

E and F are Borel bireducible. X/E and Y/F are Borel isomorphic.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Borel bireducibility vs Borel isomorphism

Theorem (Motto Ros 2012)

If E, F are Borel orbit equivalence relations of actions of Polish groups

  • n the Polish spaces X, Y, then the following are equivalent:

E and F are Borel bireducible. X/E and Y/F are Borel isomorphic.

Corollary

If G, H are countable groups, then the following are equivalent: The unitary equivalence relations ≈G and ≈H are Borel bireducible. The unitary duals Irr∞(G)/ ≈G and Irr∞(H)/ ≈H are Borel isomorphic.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Four test cases

Question

What can we say about the relative complexity of the unitary duals

  • f the following groups?

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Four test cases

Question

What can we say about the relative complexity of the unitary duals

  • f the following groups?

The free group F∞ on infinitely many generators

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Four test cases

Question

What can we say about the relative complexity of the unitary duals

  • f the following groups?

The free group F∞ on infinitely many generators SL(3, Z)

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Four test cases

Question

What can we say about the relative complexity of the unitary duals

  • f the following groups?

The free group F∞ on infinitely many generators SL(3, Z) The infinite alternating group Alt(N)

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Four test cases

Question

What can we say about the relative complexity of the unitary duals

  • f the following groups?

The free group F∞ on infinitely many generators SL(3, Z) The infinite alternating group Alt(N) The direct sum Σ of infinitely copies of Sym(3)

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Representation universal groups

Definition

A countable group G is representation universal if ≈H is Borel reducible to ≈G for every countable group H.

Observation

F∞ is representation universal.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Representation universal groups

Definition

A countable group G is representation universal if ≈H is Borel reducible to ≈G for every countable group H.

Observation

F∞ is representation universal.

Proof.

If θ : F∞ → G is a surjective homomorphism, then the induced map Irr∞(G) → Irr∞(F∞) ϕ → ϕ ◦ θ is a Borel reduction from ≈G to ≈ F∞.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Representation universal groups

Proposition

If G, H are countable groups and there exists a surjective homomorphism G → H, then ≈H is Borel reducible to ≈G.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Representation universal groups

Proposition

If G, H are countable groups and there exists a surjective homomorphism G → H, then ≈H is Borel reducible to ≈G.

Open Problem

Does the existence of an injective homomorphism G ֒ → H imply that ≈G is Borel reducible to ≈H?

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Representation universal groups

Proposition

If G, H are countable groups and there exists a surjective homomorphism G → H, then ≈H is Borel reducible to ≈G.

Open Problem

Does the existence of an injective homomorphism G ֒ → H imply that ≈G is Borel reducible to ≈H?

Open Problem

Is SL(3, Z) representation universal?

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Representation minimal groups

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Representation minimal groups

Definition

A countable group G is representation minimal if: G is not abelian-by-finite. If H is countable and not abelian-by-finite, then ≈G is Borel reducible to ≈H.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Representation minimal groups

Definition

A countable group G is representation minimal if: G is not abelian-by-finite. If H is countable and not abelian-by-finite, then ≈G is Borel reducible to ≈H. Making essential use of Elliott (1977) and Sutherland (1983)

Theorem (Thomas 2012)

If the countable group G is amenable and not abelian-by-finite, then G is representation minimal.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Amenable groups

Definition

A countable group G is amenable if there exists a left-invariant finitely additive probability measure µ : P(G) → [ 0, 1 ].

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Amenable groups

Definition

A countable group G is amenable if there exists a left-invariant finitely additive probability measure µ : P(G) → [ 0, 1 ].

The “Obvious” Examples

The class E of elementary amenable groups is the smallest collection

  • f countable groups such that:

E contains all finite groups and all countable abelian groups. If G ∈ E and H G, then H ∈ E. If G ∈ E and N G, then G/N ∈ E. If N G and N, G/N ∈ E, then G ∈ E. E is closed under countable directed limits.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Amenable groups

Some Non-Obvious Examples

(Grigorchuk 1984) Groups of intermediate growth.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Amenable groups

Some Non-Obvious Examples

(Grigorchuk 1984) Groups of intermediate growth. (Juschenko-Monod 2012) There exist infinite simple finitely generated amenable groups.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Amenable groups

Some Non-Obvious Examples

(Grigorchuk 1984) Groups of intermediate growth. (Juschenko-Monod 2012) There exist infinite simple finitely generated amenable groups.

Theorem (Thomas 2012)

If the countable group G is amenable and not abelian-by-finite, then G is representation minimal.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Amenable groups

Some Non-Obvious Examples

(Grigorchuk 1984) Groups of intermediate growth. (Juschenko-Monod 2012) There exist infinite simple finitely generated amenable groups.

Theorem (Thomas 2012)

If the countable group G is amenable and not abelian-by-finite, then G is representation minimal.

Remark

The proof ultimately depends upon the Ornstein-Weiss Theorem that if G, H are countable amenable groups, then any free ergodic measure-preserving actions of G, H are orbit equivalent.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Summing up

✉ ✉ Amenable

Abelian-by-finite “Very large groups” ??

✉

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Summing up

✉ ✉ Amenable

Abelian-by-finite “Very large groups” ??

✉

Theorem (Thomas 2012)

F2 is representation universal.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Summing up

✉ ✉ Amenable

Abelian-by-finite “Very large groups” ??

✉

Theorem (Thomas 2012)

F2 is representation universal.

Definition

A countable group G is very large if there exists a surjective homomorphism G → F2.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Summing up

✉ ✉ Amenable

Abelian-by-finite “Very large groups” ??

✉

Theorem (Thomas 2012)

F2 is representation universal.

Definition

A countable group G is very large if there exists a surjective homomorphism G → F2.

Conjecture (Thomas 2016)

Every countable non-amenable group is representation universal.

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016

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Summing up

✉ ✉ Amenable

Abelian-by-finite “Very large groups” ??

✉

Theorem (Thomas 2012)

F2 is representation universal.

Definition

A countable group G is very large if there exists a surjective homomorphism G → F2.

Conjecture (Thomas 2016)

Every countable non-amenable group is representation universal. The End

Simon Thomas (Rutgers) Lezione Lagrangiana 26th May 2016