Supramenable groups and their actions on locally compact Hausdorff - - PowerPoint PPT Presentation

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Supramenable groups and their actions on locally compact Hausdorff spaces

Mikael Rørdam rordam@math.ku.dk

Department of Mathematics University of Copenhagen

Workshop on C ∗-algebras and Noncommutative Dynamics, Sde Boker, Israel, March 11, 2013 (Joint with Julian Kellerhals and Nicolas Monod)

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Outline

1

Dynamical systems and crossed product C ∗-algebras

2

Paradoxical sets - Tarski’s theorem

3

Supramenable groups

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

X = compact or locally compact Hausdorff space, Γ = (countable) discrete group which acts on X. C0(X) ⋊red Γ is nuclear ⇐ ⇒ Γ X amenable [Anantharaman-Delaroche] Tracial states on C0(X) ⋊red Γ ↔ Γ-invariant probability measures on X C0(X) ⋊red Γ is simple ⇐ Γ X is topologically free and

• minimal. [Archbold–Spielberg]

Definition Γ X is topologically free if ∀t ∈ Γ \ {e}: {x ∈ X | t.x = x} is dense in X. Note: C ∗

red(Fn) is simple, but Fn {pt} is not topologically free

(or free).

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

X = compact or locally compact Hausdorff space, Γ = (countable) discrete group which acts on X. C0(X) ⋊red Γ is nuclear ⇐ ⇒ Γ X amenable [Anantharaman-Delaroche] Tracial states on C0(X) ⋊red Γ ↔ Γ-invariant probability measures on X C0(X) ⋊red Γ is simple ⇐ Γ X is topologically free and

• minimal. [Archbold–Spielberg]

Definition Γ X is topologically free if ∀t ∈ Γ \ {e}: {x ∈ X | t.x = x} is dense in X. Note: C ∗

red(Fn) is simple, but Fn {pt} is not topologically free

(or free).

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Definition (Anantharaman-Delaroche) An action of a countable discrete group Γ on a locally compact space X is said to be amenable if there is a net of continuous maps mi : X → Prob(Γ) (written x → mx

i ) such that

t.mx

i − mt.x i

1 → 0 uniformly on all compact subsets of X and for all t ∈ Γ. If Γ is amenable, then we can choose the mi’s above being

• constant. Hence any action of an amenable group is

amenable. If X = {pt}, then Γ acts amenably on X iff Γ is amenable. Any proper action is amenable. If X has an invariant probability measure, then Γ X is amenable if and only if Γ is amenable.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Definition (Anantharaman-Delaroche) An action of a countable discrete group Γ on a locally compact space X is said to be amenable if there is a net of continuous maps mi : X → Prob(Γ) (written x → mx

i ) such that

t.mx

i − mt.x i

1 → 0 uniformly on all compact subsets of X and for all t ∈ Γ. If Γ is amenable, then we can choose the mi’s above being

• constant. Hence any action of an amenable group is

amenable. If X = {pt}, then Γ acts amenably on X iff Γ is amenable. Any proper action is amenable. If X has an invariant probability measure, then Γ X is amenable if and only if Γ is amenable.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Definition: Γ X is regular if C0(X) ⋊full Γ = C0(X) ⋊red Γ. Anantharaman-Delaroche proved the following: Γ X amenable = ⇒ Γ X regular. Γ X amenable ⇐ ⇒ C0(X) ⋊ Γ is nuclear. Jean-Louis Tu proved: Γ X amenable = ⇒ C0(X) ⋊ Γ is in the UCT class. Example (The Roe algebra) The Roe algebra associated with a discrete group Γ is the crossed product: ℓ∞(Γ) ⋊red Γ = C(βΓ) ⋊red Γ where Γ acts on ℓ∞(Γ) by left translation. The left multiplication action Γ Γ extends (uniquely) to an action Γ βΓ. Γ βΓ is amenable ⇐ ⇒ Γ is exact. [Ozawa] Γ βΓ is free for all Γ. Γ βΓ is not minimal (unless Γ is finite).

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Definition: Γ X is regular if C0(X) ⋊full Γ = C0(X) ⋊red Γ. Anantharaman-Delaroche proved the following: Γ X amenable = ⇒ Γ X regular. Γ X amenable ⇐ ⇒ C0(X) ⋊ Γ is nuclear. Jean-Louis Tu proved: Γ X amenable = ⇒ C0(X) ⋊ Γ is in the UCT class. Example (The Roe algebra) The Roe algebra associated with a discrete group Γ is the crossed product: ℓ∞(Γ) ⋊red Γ = C(βΓ) ⋊red Γ where Γ acts on ℓ∞(Γ) by left translation. The left multiplication action Γ Γ extends (uniquely) to an action Γ βΓ. Γ βΓ is amenable ⇐ ⇒ Γ is exact. [Ozawa] Γ βΓ is free for all Γ. Γ βΓ is not minimal (unless Γ is finite).

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Definition: Γ X is regular if C0(X) ⋊full Γ = C0(X) ⋊red Γ. Anantharaman-Delaroche proved the following: Γ X amenable = ⇒ Γ X regular. Γ X amenable ⇐ ⇒ C0(X) ⋊ Γ is nuclear. Jean-Louis Tu proved: Γ X amenable = ⇒ C0(X) ⋊ Γ is in the UCT class. Example (The Roe algebra) The Roe algebra associated with a discrete group Γ is the crossed product: ℓ∞(Γ) ⋊red Γ = C(βΓ) ⋊red Γ where Γ acts on ℓ∞(Γ) by left translation. The left multiplication action Γ Γ extends (uniquely) to an action Γ βΓ. Γ βΓ is amenable ⇐ ⇒ Γ is exact. [Ozawa] Γ βΓ is free for all Γ. Γ βΓ is not minimal (unless Γ is finite).

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Recall: Γ X is regular if C0(X) ⋊full Γ = C0(X) ⋊red Γ, Γ X amenable = ⇒ Γ X regular. Γ X amenable ⇐ ⇒ C0(X) ⋊ Γ is nuclear. Proposition (Archbold-Spielberg, 1993) C0(X) ⋊full Γ is simple ⇐ ⇒ Γ X is minimal, topologically free and regular. Corollary (Anantharaman-Delaroche + Archbold-Spielberg) C0(X) ⋊red Γ is simple and nuclear ⇐ ⇒ Γ X is minimal, topologically free and amenable.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Recall: Γ X is regular if C0(X) ⋊full Γ = C0(X) ⋊red Γ, Γ X amenable = ⇒ Γ X regular. Γ X amenable ⇐ ⇒ C0(X) ⋊ Γ is nuclear. Proposition (Archbold-Spielberg, 1993) C0(X) ⋊full Γ is simple ⇐ ⇒ Γ X is minimal, topologically free and regular. Corollary (Anantharaman-Delaroche + Archbold-Spielberg) C0(X) ⋊red Γ is simple and nuclear ⇐ ⇒ Γ X is minimal, topologically free and amenable.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Recall: Γ X is regular if C0(X) ⋊full Γ = C0(X) ⋊red Γ, Γ X amenable = ⇒ Γ X regular. Γ X amenable ⇐ ⇒ C0(X) ⋊ Γ is nuclear. Proposition (Archbold-Spielberg, 1993) C0(X) ⋊full Γ is simple ⇐ ⇒ Γ X is minimal, topologically free and regular. Corollary (Anantharaman-Delaroche + Archbold-Spielberg) C0(X) ⋊red Γ is simple and nuclear ⇐ ⇒ Γ X is minimal, topologically free and amenable.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Consider now the unital case, i.e., X compact.

• C(X) ⋊red Γ nuclear =

⇒ Γ exact (C ∗

red(Γ) ⊆ C(X) ⋊red Γ).

Proposition (Anantharaman-Delaroche) Suppose C(X) ⋊red Γ is simple and nuclear. Then: Γ amenable ⇐ ⇒ C(X) ⋊red Γ is stably finite. Γ non-amenable ⇐ ⇒ ∃n : Mn(C(X) ⋊red Γ) is properly infinite. Question Suppose C(X) ⋊red Γ simple and nuclear, and Γ non-amenable. Is C(X) ⋊red Γ always properly infinite? Is C(X) ⋊red Γ always purely infinite?

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Consider now the unital case, i.e., X compact.

• C(X) ⋊red Γ nuclear =

⇒ Γ exact (C ∗

red(Γ) ⊆ C(X) ⋊red Γ).

Proposition (Anantharaman-Delaroche) Suppose C(X) ⋊red Γ is simple and nuclear. Then: Γ amenable ⇐ ⇒ C(X) ⋊red Γ is stably finite. Γ non-amenable ⇐ ⇒ ∃n : Mn(C(X) ⋊red Γ) is properly infinite. Question Suppose C(X) ⋊red Γ simple and nuclear, and Γ non-amenable. Is C(X) ⋊red Γ always properly infinite? Is C(X) ⋊red Γ always purely infinite?

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

X = metrizable, locally compact, totally disconnected space, Γ = countable discrete group, and Γ X. Proposition (Archbold-Spielberg, Laca-Spielberg) Suppose that Γ X minimal and topologically free, every non-zero projection in C0(X) is properly infinite in C0(X) ⋊red Γ. Then C0(X) ⋊red Γ is simple and purely infinite. Corollary C0(X) ⋊red Γ is a Kirchberg algebra in the UCT class if and only if Γ X is minimal, topologically free, and amenable, every non-zero projection in C0(X) is properly infinite in C0(X) ⋊red Γ.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

X = metrizable, locally compact, totally disconnected space, Γ = countable discrete group, and Γ X. Proposition (Archbold-Spielberg, Laca-Spielberg) Suppose that Γ X minimal and topologically free, every non-zero projection in C0(X) is properly infinite in C0(X) ⋊red Γ. Then C0(X) ⋊red Γ is simple and purely infinite. Corollary C0(X) ⋊red Γ is a Kirchberg algebra in the UCT class if and only if Γ X is minimal, topologically free, and amenable, every non-zero projection in C0(X) is properly infinite in C0(X) ⋊red Γ.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Definition Γ X and X totally disconnected. Let E ⊆ X be compact-open. Then E is (X, Γ)-paradoxical if ∃ compact-open pairwise disjoint subsets E1, E2, . . . , En+m of E and t1, t2, . . . , tn+m ∈ Γ st E ⊆

n

• j=1

tj.Ej, E ⊆

n+m

• j=n+1

tj.Ej. We say that Γ X is purely infinite if all non-empty compact-open subsets of X are (X, Γ)-paradoxical. Lemma Γ X, X totally disconnected, E ⊆ X compact-open. (1) There is no invariant Radon measure µ on X st 0 < µ(E) < ∞. (2) 1E is properly infinite in C0(X) ⋊red Γ. (3) E is (X, Γ)-paradoxical. Then (3) ⇒ (2) ⇒ (1).

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Definition Γ X and X totally disconnected. Let E ⊆ X be compact-open. Then E is (X, Γ)-paradoxical if ∃ compact-open pairwise disjoint subsets E1, E2, . . . , En+m of E and t1, t2, . . . , tn+m ∈ Γ st E ⊆

n

• j=1

tj.Ej, E ⊆

n+m

• j=n+1

tj.Ej. We say that Γ X is purely infinite if all non-empty compact-open subsets of X are (X, Γ)-paradoxical. Lemma Γ X, X totally disconnected, E ⊆ X compact-open. (1) There is no invariant Radon measure µ on X st 0 < µ(E) < ∞. (2) 1E is properly infinite in C0(X) ⋊red Γ. (3) E is (X, Γ)-paradoxical. Then (3) ⇒ (2) ⇒ (1).

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Corollary C0(X) ⋊red Γ is a Kirchberg algebra in the UCT class if and only if Γ X is minimal, topologically free, and amenable, every non-zero projection in C0(X) is properly infinite in C0(X) ⋊red Γ. Corollary C0(X) ⋊red Γ is a Kirchberg algebra in the UCT class if Γ X is minimal, topologically free, amenable, and purely infinite. Remark Pere Ara and Ruy Exel has recently shown that (1) (3) (in the previous lemma). They have counterexamples where Γ = F2 acts

• n the Cantor set.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Corollary C0(X) ⋊red Γ is a Kirchberg algebra in the UCT class if and only if Γ X is minimal, topologically free, and amenable, every non-zero projection in C0(X) is properly infinite in C0(X) ⋊red Γ. Corollary C0(X) ⋊red Γ is a Kirchberg algebra in the UCT class if Γ X is minimal, topologically free, amenable, and purely infinite. Remark Pere Ara and Ruy Exel has recently shown that (1) (3) (in the previous lemma). They have counterexamples where Γ = F2 acts

• n the Cantor set.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Corollary C0(X) ⋊red Γ is a Kirchberg algebra in the UCT class if and only if Γ X is minimal, topologically free, and amenable, every non-zero projection in C0(X) is properly infinite in C0(X) ⋊red Γ. Corollary C0(X) ⋊red Γ is a Kirchberg algebra in the UCT class if Γ X is minimal, topologically free, amenable, and purely infinite. Remark Pere Ara and Ruy Exel has recently shown that (1) (3) (in the previous lemma). They have counterexamples where Γ = F2 acts

• n the Cantor set.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Example The left-multiplication action Γ Γ is amenable (for all Γ), and c0(Γ) ⋊ Γ ∼ = K(ℓ2(Γ)). Example (Archbold, Kumjian) There is an action of Z2 ∗ Z3 on the Cantor set X st C(X) ⋊red (Z2 ∗ Z3) ∼ = O2. Example (Anantharaman-Delaroche) If Γ is a non-elementary hyperbolic group and ∂Γ its boundary, then C(∂Γ) ⋊red Γ is a Kirchberg algebra in the UCT class. Example (Hjorth-Molberg) Every infinite countable discrete group admits a free action on the Cantor set X with an invariant probability measure; and it also admits a free minimal action on X (not necessarily with an invariant probability measure).

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Example The left-multiplication action Γ Γ is amenable (for all Γ), and c0(Γ) ⋊ Γ ∼ = K(ℓ2(Γ)). Example (Archbold, Kumjian) There is an action of Z2 ∗ Z3 on the Cantor set X st C(X) ⋊red (Z2 ∗ Z3) ∼ = O2. Example (Anantharaman-Delaroche) If Γ is a non-elementary hyperbolic group and ∂Γ its boundary, then C(∂Γ) ⋊red Γ is a Kirchberg algebra in the UCT class. Example (Hjorth-Molberg) Every infinite countable discrete group admits a free action on the Cantor set X with an invariant probability measure; and it also admits a free minimal action on X (not necessarily with an invariant probability measure).

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Example The left-multiplication action Γ Γ is amenable (for all Γ), and c0(Γ) ⋊ Γ ∼ = K(ℓ2(Γ)). Example (Archbold, Kumjian) There is an action of Z2 ∗ Z3 on the Cantor set X st C(X) ⋊red (Z2 ∗ Z3) ∼ = O2. Example (Anantharaman-Delaroche) If Γ is a non-elementary hyperbolic group and ∂Γ its boundary, then C(∂Γ) ⋊red Γ is a Kirchberg algebra in the UCT class. Example (Hjorth-Molberg) Every infinite countable discrete group admits a free action on the Cantor set X with an invariant probability measure; and it also admits a free minimal action on X (not necessarily with an invariant probability measure).

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Outline

1

Dynamical systems and crossed product C ∗-algebras

2

Paradoxical sets - Tarski’s theorem

3

Supramenable groups

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Definition (Paradoxical sets) Let Γ be a discrete group acting on a (discrete) set X. A set E ⊆ X is said to be paradoxical if there are pairwise disjoint subsets V1, V2, . . . , Vn+m ⊆ E and t1, t2, . . . , tn+m ∈ Γ st

n

• j=1

tj.Vj =

n+m

• j=n+1

tj.Vj = E. Suppose that X = Γ and Γ acts on itself by left-multiplication. Let E ⊆ Γ. A map σ: E → Γ is said to be a congruence if there is a finite set S ⊆ Γ such that σ(t)t−1 ∈ S for all t ∈ E. ∀A ⊆ E : σ(A) ∼Γ A, when σ: E → Γ is a congruence. E ⊆ Γ is paradoxical iff there exist congruences σ± : E → E such that σ+(E) ∩ σ−(E) = ∅.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Definition (Paradoxical sets) Let Γ be a discrete group acting on a (discrete) set X. A set E ⊆ X is said to be paradoxical if there are pairwise disjoint subsets V1, V2, . . . , Vn+m ⊆ E and t1, t2, . . . , tn+m ∈ Γ st

n

• j=1

tj.Vj =

n+m

• j=n+1

tj.Vj = E. Suppose that X = Γ and Γ acts on itself by left-multiplication. Let E ⊆ Γ. A map σ: E → Γ is said to be a congruence if there is a finite set S ⊆ Γ such that σ(t)t−1 ∈ S for all t ∈ E. ∀A ⊆ E : σ(A) ∼Γ A, when σ: E → Γ is a congruence. E ⊆ Γ is paradoxical iff there exist congruences σ± : E → E such that σ+(E) ∩ σ−(E) = ∅.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Theorem (Tarski) Let Γ be a discrete group acting on a set X, and let E ⊆ X. Then there exists a Γ-invariant finitely additive measure µ on P(X) such that 0 < µ(E) < ∞ if and only if E is not Γ-paradoxical. Theorem (R.-Sierakowski) Let Γ be a countable discrete group acting on a set X (eg. X could be Γ itself). The following are equivalent for every E ⊆ X: (i) E is Γ-paradoxical. (ii) 1E is properly infinite in ℓ∞(X) ⋊red Γ. (iii) There is no lower semicontinuous tracial weight τ on ℓ∞(X) ⋊red Γ for which 0 < τ(1E) < ∞. (iv) The n-fold direct sum 1E ⊕ · · · ⊕ 1E is properly infinite in ℓ∞(X) ⋊red Γ for some n.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Theorem (Tarski) Let Γ be a discrete group acting on a set X, and let E ⊆ X. Then there exists a Γ-invariant finitely additive measure µ on P(X) such that 0 < µ(E) < ∞ if and only if E is not Γ-paradoxical. Theorem (R.-Sierakowski) Let Γ be a countable discrete group acting on a set X (eg. X could be Γ itself). The following are equivalent for every E ⊆ X: (i) E is Γ-paradoxical. (ii) 1E is properly infinite in ℓ∞(X) ⋊red Γ. (iii) There is no lower semicontinuous tracial weight τ on ℓ∞(X) ⋊red Γ for which 0 < τ(1E) < ∞. (iv) The n-fold direct sum 1E ⊕ · · · ⊕ 1E is properly infinite in ℓ∞(X) ⋊red Γ for some n.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Outline

1

Dynamical systems and crossed product C ∗-algebras

2

Paradoxical sets - Tarski’s theorem

3

Supramenable groups

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Definition (Rosenblatt, 1974) A group Γ is supramenable if for all A ⊆ Γ there exists a Γ-invariant finitely additive measure on Γ such that µ(A) = 1. Equivalenty, Γ is supramenable if Γ contains no paradoxical subsets. Rosenblatt proved the following: All abelian groups are supramenable. Every finitely generated group of subexponential growth (in particular, of polynomial growth) is supramenable. Any group that contains a free semigroup with two (or more) generators is not supramenable. In general: bad permanence properties. Not known if G, H supramenable = ⇒ G × H supramenable.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Definition (Rosenblatt, 1974) A group Γ is supramenable if for all A ⊆ Γ there exists a Γ-invariant finitely additive measure on Γ such that µ(A) = 1. Equivalenty, Γ is supramenable if Γ contains no paradoxical subsets. Rosenblatt proved the following: All abelian groups are supramenable. Every finitely generated group of subexponential growth (in particular, of polynomial growth) is supramenable. Any group that contains a free semigroup with two (or more) generators is not supramenable. In general: bad permanence properties. Not known if G, H supramenable = ⇒ G × H supramenable.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Theorem (Rosenblatt) Let Γ be a solvable group. TFAE:

1 Γ is supramenable, 2 Γ has sub-exponential growth, 3 Γ does not contain a free semigroup (of two or more

generators),

4 Γ is essentially nilpotent.

Well-known fact: A group Γ is amenable if and only if whenever it acts on a compact Hausdorff space X, then X has a Γ-invariant probability measure. An obervation (by Monod) that is essential for this work: A group Γ is supramenable if and only if whenever it acts co-compactly on a locally compact Hausdorff space X, then X has a non-zero Γ-invariant Radon measure.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Theorem (Rosenblatt) Let Γ be a solvable group. TFAE:

1 Γ is supramenable, 2 Γ has sub-exponential growth, 3 Γ does not contain a free semigroup (of two or more

generators),

4 Γ is essentially nilpotent.

Well-known fact: A group Γ is amenable if and only if whenever it acts on a compact Hausdorff space X, then X has a Γ-invariant probability measure. An obervation (by Monod) that is essential for this work: A group Γ is supramenable if and only if whenever it acts co-compactly on a locally compact Hausdorff space X, then X has a non-zero Γ-invariant Radon measure.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Example The ax + b group (over Q) is not supramenable. The Thompson group F is not supramenable. BS(1, m) = a, b | bab−1 = am (with m = ±1) is solvable, finitely generated, and non-supramenable (it contains a free semigroup: b2, b2a).

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Supramenability and the Roe algebra

Fact: E ⊆ Γ is paradoxical iff 1E is a properly infinite projection in ℓ∞(Γ) ⋊red Γ. One can use this fact to prove: Proposition Let Γ be a discrete group. Then Γ is supramenable if and only if the Roe algebra ℓ∞(Γ) ⋊red Γ contains no properly infinite projections. Γ is non-amenable ⇐ ⇒ Γ is paradoxical ⇐ ⇒ ℓ∞(Γ) ⋊red Γ is properly infinite. Γ non-supramenable = ⇒ ℓ∞(Γ) ⋊red Γ is infinite (by the Proposition above). If Γ contains an element of infinite order, then ℓ∞(Γ) ⋊red Γ is infinite. If Γ is locally finite, then ℓ∞(Γ) ⋊red Γ is finite. Question: ℓ∞(Γ) ⋊red Γ is finite ⇐ ⇒ Γ locally finite?

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Supramenability and the Roe algebra

Fact: E ⊆ Γ is paradoxical iff 1E is a properly infinite projection in ℓ∞(Γ) ⋊red Γ. One can use this fact to prove: Proposition Let Γ be a discrete group. Then Γ is supramenable if and only if the Roe algebra ℓ∞(Γ) ⋊red Γ contains no properly infinite projections. Γ is non-amenable ⇐ ⇒ Γ is paradoxical ⇐ ⇒ ℓ∞(Γ) ⋊red Γ is properly infinite. Γ non-supramenable = ⇒ ℓ∞(Γ) ⋊red Γ is infinite (by the Proposition above). If Γ contains an element of infinite order, then ℓ∞(Γ) ⋊red Γ is infinite. If Γ is locally finite, then ℓ∞(Γ) ⋊red Γ is finite. Question: ℓ∞(Γ) ⋊red Γ is finite ⇐ ⇒ Γ locally finite?

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Our main result

Y = locally compact non-compact Cantor set. Theorem Let Γ be a countable exact group. Then the following are equivalent: (1) Γ is not supramenable. (2) Γ admits a free, minimal, amenable, purely infinite action on Y . If Γ Y is as in (2), then C0(Y ) ⋊ Γ is a stable Kirchberg algebra in the UCT class. Note: Γ Y amenable Γ exact.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Construction of actions of groups on locally compact spaces

Begin with Γ βΓ. Recall that this action is free; and amenable if Γ is exact. Take A ⊆ Γ and put K = ¯ A ⊆ βΓ. Note that K is compact-open in βΓ. Put XA =

• t∈Γ

t.K ⊆ βΓ. Then Γ XA, the action is co-compact, and XA is locally compact. Claim: XA admits a non-zero Γ-invariant Radon measure iff A is non-paradoxical. This shows that: if all co-compact actions of Γ on any locally compact Hausdorff space admit a non-zero invariant Radon measure, then Γ must be supramenable.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Construction of actions of groups on locally compact spaces

Begin with Γ βΓ. Recall that this action is free; and amenable if Γ is exact. Take A ⊆ Γ and put K = ¯ A ⊆ βΓ. Note that K is compact-open in βΓ. Put XA =

• t∈Γ

t.K ⊆ βΓ. Then Γ XA, the action is co-compact, and XA is locally compact. Claim: XA admits a non-zero Γ-invariant Radon measure iff A is non-paradoxical. This shows that: if all co-compact actions of Γ on any locally compact Hausdorff space admit a non-zero invariant Radon measure, then Γ must be supramenable.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Construction of actions of groups on locally compact spaces, ctd.

Claim: XA admits a non-zero Γ-invariant Radon measure iff A is non-paradoxical. Proof: (Only if.) Suppose that λ is a non-zero invariant Radon measure on XA. Recall K = ¯ A ⊆ βΓ is compact-open. Then 0 < λ(K) < ∞. Put ΩA = {E ⊆ Γ : E ⊆

• t∈S

sA for some finite S ⊆ Γ}. If E ∈ ΩA, then ¯ E ⊆ XA. Define µ: P(Γ) → [0, ∞] by µ(E) =

• λ( ¯

E), E ∈ ΩA, ∞, E / ∈ ΩA, E ⊆ Γ. Then µ is a finitely additive Γ-invariant measure, and µ(A) = λ(K). Hence A is non-paradoxical.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Construction of actions of groups on locally compact spaces, ctd.

A ⊆ Γ Γ XA. The action is free and co-compact. It is amenable if Γ is exact. Since XA =

t∈Γ t.K, where K = ¯

A, and K is compact, it follows that XA has a maximal proper Γ-invariant open subset U. Proposition Put Z = XA \ U ⊆ βΓ. Then Z is Γ-invariant, locally compact, and totally disconnected. Γ Z is free and minimal (and amenable if Γ is exact). If A is paradoxical, then Γ Z is purely infinite. Def: Write A ≪ Γ if A Γ Γ \

s∈S sA for all finite S ⊆ Γ.

Fact: If A ≪ Γ, then Z cannot be compact. Fact: If Γ infinite, then ∃ A ≪ Γ st A infinite. Thm: If Γ non-amenable, then ∃ A ≪ Γ st A paradoxical.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Construction of actions of groups on locally compact spaces, ctd.

A ⊆ Γ Γ XA. The action is free and co-compact. It is amenable if Γ is exact. Since XA =

t∈Γ t.K, where K = ¯

A, and K is compact, it follows that XA has a maximal proper Γ-invariant open subset U. Proposition Put Z = XA \ U ⊆ βΓ. Then Z is Γ-invariant, locally compact, and totally disconnected. Γ Z is free and minimal (and amenable if Γ is exact). If A is paradoxical, then Γ Z is purely infinite. Def: Write A ≪ Γ if A Γ Γ \

s∈S sA for all finite S ⊆ Γ.

Fact: If A ≪ Γ, then Z cannot be compact. Fact: If Γ infinite, then ∃ A ≪ Γ st A infinite. Thm: If Γ non-amenable, then ∃ A ≪ Γ st A paradoxical.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Construction of actions of groups on locally compact spaces, ctd.

A ⊆ Γ Γ XA. The action is free and co-compact. It is amenable if Γ is exact. Since XA =

t∈Γ t.K, where K = ¯

A, and K is compact, it follows that XA has a maximal proper Γ-invariant open subset U. Proposition Put Z = XA \ U ⊆ βΓ. Then Z is Γ-invariant, locally compact, and totally disconnected. Γ Z is free and minimal (and amenable if Γ is exact). If A is paradoxical, then Γ Z is purely infinite. Def: Write A ≪ Γ if A Γ Γ \

s∈S sA for all finite S ⊆ Γ.

Fact: If A ≪ Γ, then Z cannot be compact. Fact: If Γ infinite, then ∃ A ≪ Γ st A infinite. Thm: If Γ non-amenable, then ∃ A ≪ Γ st A paradoxical.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Construction of actions of groups ... , ctd.

Proposition Put Z = XA \ U ⊆ βΓ. Then Z is Γ-invariant, locally compact, and totally disconnected. Γ Z is free and minimal (and amenable if Γ is exact). If A is paradoxical, then Γ Z is purely infinite. We want Z to be non-compact and without isolated points. If A ≪ Γ, then Z cannot be compact. Can Z have isolated points? Only if Z = Γ, and this will for sure happen if A is finite (then XA = Γ). It will not happen if A is paradoxical (no isolated point can be paradoxical). If A satisfies |A ∩ sA| < ∞ for all e = s ∈ Γ, then Z has isolated

• points. Moreover, if A is infinite with this property, then

1A (ℓ∞(Γ) ⋊red Γ) 1A has a character! All infinite groups contains an infinite set A with this property.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Construction of actions of groups ... , ctd.

Proposition Put Z = XA \ U ⊆ βΓ. Then Z is Γ-invariant, locally compact, and totally disconnected. Γ Z is free and minimal (and amenable if Γ is exact). If A is paradoxical, then Γ Z is purely infinite. We want Z to be non-compact and without isolated points. If A ≪ Γ, then Z cannot be compact. Can Z have isolated points? Only if Z = Γ, and this will for sure happen if A is finite (then XA = Γ). It will not happen if A is paradoxical (no isolated point can be paradoxical). If A satisfies |A ∩ sA| < ∞ for all e = s ∈ Γ, then Z has isolated

• points. Moreover, if A is infinite with this property, then

1A (ℓ∞(Γ) ⋊red Γ) 1A has a character! All infinite groups contains an infinite set A with this property.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Construction of actions of groups ... , ctd.

Proposition Put Z = XA \ U ⊆ βΓ. Then Z is Γ-invariant, locally compact, and totally disconnected. Γ Z is free and minimal (and amenable if Γ is exact). If A is paradoxical, then Γ Z is purely infinite. We want Z to be non-compact and without isolated points. If A ≪ Γ, then Z cannot be compact. Can Z have isolated points? Only if Z = Γ, and this will for sure happen if A is finite (then XA = Γ). It will not happen if A is paradoxical (no isolated point can be paradoxical). If A satisfies |A ∩ sA| < ∞ for all e = s ∈ Γ, then Z has isolated

• points. Moreover, if A is infinite with this property, then

1A (ℓ∞(Γ) ⋊red Γ) 1A has a character! All infinite groups contains an infinite set A with this property.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Definition Let Γ and Λ be discrete groups. A map f : Λ → Γ is said to be Lipschitz if for every finite F ⊆ Λ there is a finite F ′ ⊆ Γ st ∀s, t ∈ Λ : st−1 ∈ F = ⇒ f (s)f (t)−1 ∈ F ′. f is said to be a uniform map if it is Lipschitz and satisfies that for every finite F ′ ⊆ Γ there is a finite F ⊆ Λ st ∀s, t ∈ Λ : f (s)f (t)−1 ∈ F ′ = ⇒ st−1 ∈ F. f is said to be a uniform embedding if it is an injective uniform map. Any group homomorphism is Lipschitz. Any group homomorphism with finite kernel is a uniform map. Def: An injective map σ: A ⊆ Γ → Γ for which there is a finite S ⊆ Γ st σ(t)t−1 ∈ S for all t ∈ A is called a piecewise congruence. Fact: A ⊆ Γ is paradoxical iff there are piecewise congruences σ± : A → A such that σ+(A) ∩ σ−(A) = ∅.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Definition Let Γ and Λ be discrete groups. A map f : Λ → Γ is said to be Lipschitz if for every finite F ⊆ Λ there is a finite F ′ ⊆ Γ st ∀s, t ∈ Λ : st−1 ∈ F = ⇒ f (s)f (t)−1 ∈ F ′. f is said to be a uniform map if it is Lipschitz and satisfies that for every finite F ′ ⊆ Γ there is a finite F ⊆ Λ st ∀s, t ∈ Λ : f (s)f (t)−1 ∈ F ′ = ⇒ st−1 ∈ F. f is said to be a uniform embedding if it is an injective uniform map. Any group homomorphism is Lipschitz. Any group homomorphism with finite kernel is a uniform map. Def: An injective map σ: A ⊆ Γ → Γ for which there is a finite S ⊆ Γ st σ(t)t−1 ∈ S for all t ∈ A is called a piecewise congruence. Fact: A ⊆ Γ is paradoxical iff there are piecewise congruences σ± : A → A such that σ+(A) ∩ σ−(A) = ∅.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Definition Let Γ and Λ be discrete groups. A map f : Λ → Γ is said to be Lipschitz if for every finite F ⊆ Λ there is a finite F ′ ⊆ Γ st ∀s, t ∈ Λ : st−1 ∈ F = ⇒ f (s)f (t)−1 ∈ F ′. f is said to be a uniform map if it is Lipschitz and satisfies that for every finite F ′ ⊆ Γ there is a finite F ⊆ Λ st ∀s, t ∈ Λ : f (s)f (t)−1 ∈ F ′ = ⇒ st−1 ∈ F. f is said to be a uniform embedding if it is an injective uniform map. Any group homomorphism is Lipschitz. Any group homomorphism with finite kernel is a uniform map. Def: An injective map σ: A ⊆ Γ → Γ for which there is a finite S ⊆ Γ st σ(t)t−1 ∈ S for all t ∈ A is called a piecewise congruence. Fact: A ⊆ Γ is paradoxical iff there are piecewise congruences σ± : A → A such that σ+(A) ∩ σ−(A) = ∅.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

If A ⊆ Λ, if f : Λ → Γ is injective Lipschitz, and if σ: A → Λ is a piecewise congruence, then τ : f (A) → Γ given by τ ◦ f = f ◦ σ is a congruence: A

f

• σ
• f (A)

τ

• Λ

f

Γ

Any injective Lipschitz function maps paradoxical sets to paradoxical sets. Proposition Γ is non-supramenable ⇐ ⇒ ∃ injective Lipschitz map f : F2 → Γ. Proof: ”⇐”: f (F2) is a paradoxical set in Γ. ”⇒”: Construct a free binary tree in the Cayley graph of Γ from piecewise congruences σ± : A → Γ that witness the paradoxicality

• f A ⊆ Γ.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

If A ⊆ Λ, if f : Λ → Γ is injective Lipschitz, and if σ: A → Λ is a piecewise congruence, then τ : f (A) → Γ given by τ ◦ f = f ◦ σ is a congruence: A

f

• σ
• f (A)

τ

• Λ

f

Γ

Any injective Lipschitz function maps paradoxical sets to paradoxical sets. Proposition Γ is non-supramenable ⇐ ⇒ ∃ injective Lipschitz map f : F2 → Γ. Proof: ”⇐”: f (F2) is a paradoxical set in Γ. ”⇒”: Construct a free binary tree in the Cayley graph of Γ from piecewise congruences σ± : A → Γ that witness the paradoxicality

• f A ⊆ Γ.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Theorem (Benjamini-Schramm, 1997) If Γ is non-amenable = ⇒ ∃ uniform embedding f : F2 → Γ. (The existence of A ≪ Γ st A is paradoxical for every non-amenable Γ follows from this theorem of Benjamini and Schramm.) Recall: Γ is non-supramenable ⇐ ⇒ ∃ injective Lipschitz map f : F2 → Γ. {groups of sub-exponential growth} ⊆ {supramenable groups} = {groups with no Lipschitz embeddig of F2} ⊆ {groups with no uniform embedding of F2} ⊂ {amenable groups} It is not known if the two ”⊆” are strict. Cornulier and Tessera proved that any finitely generated solvable group with exponential growth admits a uniform embedding of F2. Hence the last inclusion is strict.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Theorem (Benjamini-Schramm, 1997) If Γ is non-amenable = ⇒ ∃ uniform embedding f : F2 → Γ. (The existence of A ≪ Γ st A is paradoxical for every non-amenable Γ follows from this theorem of Benjamini and Schramm.) Recall: Γ is non-supramenable ⇐ ⇒ ∃ injective Lipschitz map f : F2 → Γ. {groups of sub-exponential growth} ⊆ {supramenable groups} = {groups with no Lipschitz embeddig of F2} ⊆ {groups with no uniform embedding of F2} ⊂ {amenable groups} It is not known if the two ”⊆” are strict. Cornulier and Tessera proved that any finitely generated solvable group with exponential growth admits a uniform embedding of F2. Hence the last inclusion is strict.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Theorem (Benjamini-Schramm, 1997) If Γ is non-amenable = ⇒ ∃ uniform embedding f : F2 → Γ. (The existence of A ≪ Γ st A is paradoxical for every non-amenable Γ follows from this theorem of Benjamini and Schramm.) Recall: Γ is non-supramenable ⇐ ⇒ ∃ injective Lipschitz map f : F2 → Γ. {groups of sub-exponential growth} ⊆ {supramenable groups} = {groups with no Lipschitz embeddig of F2} ⊆ {groups with no uniform embedding of F2} ⊂ {amenable groups} It is not known if the two ”⊆” are strict. Cornulier and Tessera proved that any finitely generated solvable group with exponential growth admits a uniform embedding of F2. Hence the last inclusion is strict.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Y = locally compact non-compact Cantor set. Question

1 Which countable discrete groups Γ admit a free, minimal,

amenable action on Y ?

2 Which countable discrete groups Γ admit a free, minimal,

amenable action on Y with a non-zero invariant Radon measure?

3 Which countable discrete groups Γ admit a free, minimal,

amenable, purely infinite action on Y ?

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Y = locally compact non-compact Cantor set. Question

1 Which countable discrete groups Γ admit a free, minimal,

amenable action on Y ?

2 Which countable discrete groups Γ admit a free, minimal,

amenable action on Y with a non-zero invariant Radon measure?

3 Which countable discrete groups Γ admit a free, minimal,

amenable, purely infinite action on Y ? If Γ Y is as in (1), then C0(Y ) ⋊ Γ is simple, nuclear and in the UCT class. If Γ Y is as in (2), then C0(Y ) ⋊ Γ is simple, nuclear, stably finite and in the UCT class. If Γ Y is as in (3), then C0(Y ) ⋊ Γ is a Kirchberg algebra in the UCT class.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Y = locally compact non-compact Cantor set. Question

1 Which countable discrete groups Γ admit a free, minimal,

amenable action on Y ?

2 Which countable discrete groups Γ admit a free, minimal,

amenable action on Y with a non-zero invariant Radon measure?

3 Which countable discrete groups Γ admit a free, minimal,

amenable, purely infinite action on Y ?

1 All groups that contain an infinite exact group as a subgroup.

Perhaps all countable infinite groups.

2 All groups that contain an infinite amenable group as a

• subgroup. Perhaps all countable infinite groups.

3 All groups that contain an exact non-supramenable group as a

• subgroup. Perhaps all non-supramenable groups.

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Induced actions

Let Γ0 be a subgroup of Γ. Then any action Γ0 X induces an action Γ Y , where Y = (Γ × X)/Γ0. If Γ0 X is minimal, then so is Γ Y . If Γ0 X is free, then so is Γ Y . If Γ0 X is amenable, then so is Γ Y . If X is the (compact) Cantor set, and if |Γ : Γ0| = ∞, then Y is the locally compact non-compact Cantor set. If X is the locally compact non-compact Cantor set, then so is Y .

nat-logo Dynamical systems and crossed product C∗-algebras Paradoxical sets - Tarski’s theorem Supramenable groups

Induced actions

Let Γ0 be a subgroup of Γ. Then any action Γ0 X induces an action Γ Y , where Y = (Γ × X)/Γ0. If Γ0 X is minimal, then so is Γ Y . If Γ0 X is free, then so is Γ Y . If Γ0 X is amenable, then so is Γ Y . If X is the (compact) Cantor set, and if |Γ : Γ0| = ∞, then Y is the locally compact non-compact Cantor set. If X is the locally compact non-compact Cantor set, then so is Y .