banach-tarski paradox and amenabilitywrobel/gigem2018/talks/... · 2018-03-03 · banach-tarski...

Banach-Tarski Paradox Amenability Orbit Equivalence

Banach-Tarski Paradox and Amenability

Konrad Wrobel

Texas A&M UniversityDepartment of Mathematics

GIG’EM 2018

March 4, 2018

Banach-Tarski

What is Banach Tarski

Start with a sphere

Partition it into finitely many sets

Rotate and translate these sets

Get two spheres of the same volume as the first

Some Quick Observations/History

Let S2 be the sphere. Let (Ai )ni=1 be a partition of S that allows

the Banach-Tarski Paradox to occur.

At least one of these subsets Ai has to be nonmeasurable.

The standard tool we use for constructing nonmeasurable setsis the Axiom of Choice(although we don’t require the fullpower of AC).

This was one of the reasons Axiom of Choice wascontroversial in the early 20th century.

Construction

Let G be a group and A an arbitrary set.Fix a group action G y A.

Paradoxicality

The set A is G -paradoxical if there exist disjoint subsetsA1, ...,An,B1, ...,Bm of G and elements a1, ..., an, b1, ..., bm of Gsuch that A =

⋃i ai · Ai =

⋃j bj · Bj .

This should look similar to the statement of Banach-Tarski, sinceBanach-Tarski can be restated as the sphere beingSO(3)-paradoxical.

Construction

Paradoxicality of F2 = 〈a, b〉For a generator g ∈ {a, a−1, b, b−1}, set S(g) to be the set of allwords that begin with the letter g .

Notice that a · S(a−1) = {e} t S(a−1) t S(b) t S(b−1)

So F2 is F2-paradoxical.

Construction

Paradoxicality of F2 = 〈a, b〉For a generator g ∈ {a, a−1, b, b−1}, set S(g) to be the set of allwords that begin with the letter g .Notice that a · S(a−1) = {e} t S(a−1) t S(b) t S(b−1)

Construction

Paradoxicality of F2 = 〈a, b〉For a generator g ∈ {a, a−1, b, b−1}, set S(g) to be the set of allwords that begin with the letter g .Notice that a · S(a−1) = {e} t S(a−1) t S(b) t S(b−1)

Construction

An action of F2 on the sphere S2

Consider the matrices

−2√2

0 −√23

These are elements of SO(3) that generate a copy of F2.

Now we have an action of F2 on the sphere S2 by rotations. Eachelement g ∈ F2 stabilizes only two points on the sphere.

So, in fact, we have a free action F 2 y S2 \ C where C is acountable set.

Construction

−2√2

0 −√23

Construction

−2√2

0 −√23

Construction

Pushing forward the paradoxicality of F 2

Recall F2 = S(a) t a · S(a−1) = S(b) t b · S(b−1).

Use Axiom of Choice to choose a single point in every orbit.Think of this set of points as the identity of F2.

Now simultaneously find sets T (a),T (a−1),T (b),T (b−1)analogously to S(a), ...

S2 \ C = T (a) t a · T (a−1) = T (b) t b · T (b−1).

So, S2 \ C is, in fact, F 2-paradoxical.

Definitions

Amenability

A countable discrete group G is amenable if there exists asequence of nonempty finite sets (Fn)n∈N such that for every g ∈ G

|Fn∆(g · Fn)||Fn|

→ 0 as n→∞

Such a sequence is called a Følner sequence.

Examples

finite groups

abelian groups

solvable groups

Some implications

G amenable=⇒ ∃ left-invariant finitely-additive probability measure on G

Proof. Take a Følner sequence (Fn) for G and fix an ultrafilter Uon N. Let A ⊆ G . Define m(A) := limU

|A∩Fn||Fn| .

G amenable =⇒ G is not paradoxicalProof. Take a left-invariant finitely additive probability measure mon G . If G were paradoxical, thenG =

⊔Ai t

⊔Bj =

⊔ai · Ai =

⊔bj · Bj . But we now get that

1 = m(G ) = m(G ) + m(G ) = 2 due to the properties of m.

Other directions are also true but not as easy.

Some implications

G amenable=⇒ ∃ left-invariant finitely-additive probability measure on GProof. Take a Følner sequence (Fn) for G and fix an ultrafilter Uon N. Let A ⊆ G . Define m(A) := limU

|A∩Fn||Fn| .

⊔Ai t

⊔Bj =

⊔ai · Ai =

Some implications

|A∩Fn||Fn| .

G amenable =⇒ G is not paradoxical

Proof. Take a left-invariant finitely additive probability measure mon G . If G were paradoxical, thenG =

⊔Ai t

⊔Bj =

⊔ai · Ai =

Some implications

|A∩Fn||Fn| .

⊔Ai t

⊔Bj =

⊔ai · Ai =

Some implications

|A∩Fn||Fn| .

⊔Ai t

⊔Bj =

⊔ai · Ai =

Ergodicity

Probability Measure Preserving

A group action on a standard measure space G y ([0, 1], λ) isprobability measure preserving(p.m.p.) if for every g ∈ G andmeasurable A ⊆ X , the measure λ(A) = λ(g · A).

Ergodic

A p.m.p. group action G y ([0, 1], λ) is ergodic if for everyG-invariant set A ⊆ X , λ(A) = 10 or λ(A) = 1.

Example Irrational translation(fix q ∈ R \Q) by Z,

n · x 7→ x + nq mod 1

Ergodicity

Probability Measure Preserving

A group action on a standard measure space G y ([0, 1], λ) isprobability measure preserving(p.m.p.) if for every g ∈ G andmeasurable A ⊆ X , the measure λ(A) = λ(g · A).

Ergodic

A p.m.p. group action G y ([0, 1], λ) is ergodic if for everyG-invariant set A ⊆ X , λ(A) = 10 or λ(A) = 1.

Example Irrational translation(fix q ∈ R \Q) by Z,

n · x 7→ x + nq mod 1

Orbit Equivalence

Two group actions on a standard measure space G ,H y ([0, 1], λ)are Orbit Equivalent if there is a measure space isomorphismf : ([0, 1], λ)→ ([0, 1], λ) such that

f (G · x) = H · f (x) for almost every x ∈ [0, 1]

Dye’s Theorem

If G ,H are amenable, then any 2 free p.m.p. ergodic actions on astandard measure space are orbit equivalent.

So, up to orbit equivalence, there is one action of amenable groups.

Orbit Equivalence

Two group actions on a standard measure space G ,H y ([0, 1], λ)are Orbit Equivalent if there is a measure space isomorphismf : ([0, 1], λ)→ ([0, 1], λ) such that

f (G · x) = H · f (x) for almost every x ∈ [0, 1]

Dye’s Theorem

If G ,H are amenable, then any 2 free p.m.p. ergodic actions on astandard measure space are orbit equivalent.

So, up to orbit equivalence, there is one action of amenable groups.

What about nonamenable groups?

Hjorth 2005

Every nonamenable group has at least 2 free p.m.p ergodicnon-orbit equivalent actions.

Epstein 2007

Every nonamenable group has at least c ≥ 2ℵ0 free p.m.p ergodicnon-orbit equivalent actions.

So, nonamenable groups have a lot of actions.

What about nonamenable groups?

Hjorth 2005

Every nonamenable group has at least 2 free p.m.p ergodicnon-orbit equivalent actions.

Epstein 2007

Every nonamenable group has at least c ≥ 2ℵ0 free p.m.p ergodicnon-orbit equivalent actions.

So, nonamenable groups have a lot of actions.

banach-tarski paradox and amenabilitywrobel/gigem2018/talks/... · 2018-03-03 · banach-tarski...

Documents