Amenable actions of the infinite permutation group Lecture III - PowerPoint PPT Presentation

Amenable actions of the infinite permutation group — Lecture III Juris Stepr¯ ans York University Young Set Theorists Meeting — March 2011, Bonn Juris Stepr¯ ans Amenable actions

It will be shown in Lecture III that if the natural action of G on N has a unique invariant mean µ then this mean is defined by µ ( A ) < r for any rational r if and only if ( ∃ Z ∈ [ G ] < ℵ 0 )( ∀ k ∈ N ) | { z ∈ Z | zk ∈ A } | < r | Z | In the case of a { 0 , 1 } -valued invariant mean µ this yields that { A ⊆ N | µ ( A ) = 1 } is an ultrafilter. The preceding definition shows that if the definition of G is simple, then so is the quantifier ” ∃ Z ∈ [ G ] < ℵ 0 ”. This ultrafilter would then have to be analytic. Juris Stepr¯ ans Amenable actions

Recall from Lecture I that the argument establishing there are no analytic subgroups of S ( ω ) that act with a unique mean relied on the fact that a unique mean, if it exists, has a nice definition. This will now be proved. Definition Let G be subgroup of S ( ω ) . A set X ⊆ ω is said to be r-thick (with respect to G) if and only if for every finite subset H ⊆ G there is n ∈ ω such that | { h ∈ H | hn ∈ X } | ≥ r | H | Juris Stepr¯ ans Amenable actions

Lemma (Wang) If G is an amenable subgroup of S ( ω ) then X ⊆ ω is r-thick if and only if there is a G-invariant mean µ on ω such that µ ( X ) ≥ r. To see this first assume that X ⊆ ω is r -thick. Using that G is amenable — and hence satisfies the Følner condition — let { F ǫ, H } ǫ> 0 , H ∈ [ G ] < ℵ 0 be such that H ⊆ F ǫ, H ∈ [ G ] < ℵ 0 if ǫ < δ and H ⊇ D then F ǫ, H ⊇ F δ, D | hF ǫ, H ∆ F ǫ, H | < ǫ for all h ∈ H . | F ǫ, H | Using the fact that X is r -thick choose for each H ∈ [ G ] < ℵ 0 and ǫ > 0 there is an integer N ǫ, H such that | { h ∈ F ǫ, H | hN ǫ, H ∈ X |} ≥ r | F ǫ, H | Juris Stepr¯ ans Amenable actions

Now define a measure µ ǫ, H by defining µ ǫ, H ( Y ) = | { h ∈ F ǫ, H | hN ǫ, H ∈ Y } | | F ǫ, H | and note that µ ǫ, H ( X ) ≥ r for all H and ǫ . Moreover, by the Følner property it follows that µ ǫ, H ( gY ) µ ǫ, H ( Y ) = � h ∈ g − 1 F ǫ, H | hN ǫ, H ∈ Y � | { h ∈ F ǫ, H | hN ǫ, H ∈ Y } | = | | | { h ∈ F ǫ, H | hN ǫ, H ∈ gY } | | { h ∈ F ǫ, H | hN ǫ, H ∈ Y } | for each g ∈ H and since | g − 1 F ǫ, H ∆ F ǫ, H | < ǫ it follows that | F ǫ, H | µ ǫ, H ( gY ) lim µ ǫ, H ( Y ) = 1 ǫ → 0 Juris Stepr¯ ans Amenable actions

Let µ be a weak ∗ limit of the µ ǫ, H along the net of ( ǫ, H ) in (0 , ∞ ) × [ G ] < ℵ 0 . This yields a G invariant measure such that µ ( X ) ≥ r . To check the other direction suppose that X ⊆ ω and that µ is a mean such that µ ( X ) ≥ r . Then let ψ : ℓ ∞ → R be the linear function defined by Lebesgue integration with respect to µ . Then for any finite H ⊆ G by linearity and G -invariance of ψ it follows that �� � � ψ χ h − 1 X = ψ ( χ h − 1 X )) = | H | µ ( X ) ≥ | H | r h ∈ H h ∈ H By the positivity of ψ this means that there must be at least one n ∈ ω such that � h ∈ H χ h − 1 X ( n ) ≥ | H | r . In other words, | { h ∈ H | hn ∈ X } | ≥ | H | r as required. Juris Stepr¯ ans Amenable actions

Definition For any group G acting on ω define a function m G on the power set of ω by m G ( X ) = sup( { r ∈ R | X is r-thick } ) . Corollary If G is an amenable group acting on ω then m G is a finitely additive probability measure if and only the action of G on ω has a unique invariant mean. Note that the preceding lemma yields the following alternate definition of m G : m G ( X ) = sup( { r ∈ R | ( ∃ µ ) µ is an invariant mean and µ ( X ) = r } ) and if there is a unique invariant mean µ this yields that m G ( X ) = µ ( X ). Hence m G is an invariant probability measure. Juris Stepr¯ ans Amenable actions

For the other direction, suppose that m G is an invariant mean. From the definition of m G it follows that if µ is an other invariant mean then µ ( X ) ≤ sup( { µ ( X ) | µ is an invariant mean } ) = m G ( X ) for every X . But if µ ( X ) � m G ( X ) for some X then µ ( ω \ X ) ≤ m G ( ω \ X ) and hence µ ( ω ) = µ ( X ) + µ ( ω \ X ) � m G ( X ) + m G ( ω \ X ) = 1. Juris Stepr¯ ans Amenable actions

Foreman showed that in the model obtained by adding ℵ 2 Cohen reals to a model of CH that there is no locally finite subgroup of S ( ω ) that acts on ω with a unique invariant mean. An analysis of his argument will show that he actually proved the following. Theorem Let P = � ξ ∈ ω 2 P ξ be a finite support product of ccc partial orders. If G ⊆ � ξ ∈ ω 2 P ξ is generic over V then in V [ G ] the following holds: There is no subgroup G ⊆ S ( ω ) acting with a unique invariant mean on ω such that for any finite set H ⊆ G there is a recursive function F H : ω → ω such that the orbit of each n under the subgroup generated by H has cardinality bounded by F H ( n ) . Note that if G is locally finite then F H is a constant function for each H . ”Recursive” is actually weaker than needed since it will be shown that F H can not be chosen from V . Juris Stepr¯ ans Amenable actions

The support of P adds ℵ 2 Cohen reals; but, for notational convenience, assume that each P ξ has exactly two maximal elements, 0 ξ and 1 ξ , and let c ξ ⊆ ω be defined by n ∈ c ξ if and only if 1 ξ + n ∈ G . Now assume that G is a P name for a subgroup G ⊆ S ( ω ) acting with a unique mean on ω such that for any finite set H ⊆ G there is a recursive function F H : ω → ω such that for each n the orbit of n under the subgroup generated by H has cardinality bounded by F H ( n ). It must be that the unique mean is m G = sup( { r ∈ R | X is r -thick } ) Juris Stepr¯ ans Amenable actions

By symmetry, there is no harm in assuming that m G ( c ξ ) < 1 for ℵ 2 of the ξ . In other words, ℵ 2 of the c ξ are not 1-thick and hence there are finite H ξ ⊆ G such that for all n ∈ ω H ξ n �⊆ c ξ Now let S ξ be a countable subset of ω 2 such that c ξ and H ξ have � η ∈ S ξ P η names. Let R be a countable set and ξ � = η be such that { ξ + j } j ∈ ω ⊆ S ξ \ R and { η + j } j ∈ ω ⊆ S η \ R . Let G R ⊆ � ρ ∈ R P ρ be generic over V . Let H ξ / G R = H ′ ξ and H η / G R = H ′ η be names in V [ G R ]. Let Q ξ = � α ∈ S ξ \ R P α and Q η = � α ∈ S η \ R P α and Q = � ρ ∈ ω 2 \ R P ρ Juris Stepr¯ ans Amenable actions

In V [ G R ] choose a condition q ∈ Q such that η = ˇ q � Q “ F H ′ F ” ξ ∪ H ′ Claim For p ≤ q the set of n ∈ ω such that � p ↾ S ξ � � Q ξ “m / ∈ � H ′ � � � | m ∈ ω ξ � n” | < ℵ 0 is finite where � H ′ ξ � is the subgroup generated by H ′ ξ . Same for η . To see this let S be the support of p and S ∗ = { j | ξ + j ∈ S } and suppose, heading towards a contradiction, that � | � p � � Q “ m / � � ∈ � H ′ � � � � Z ⊆ n ∈ ω m ∈ ω ξ � n ” | < ℵ 0 is such that | Z | > � j ∈ S ∗ F ( j ). Juris Stepr¯ ans Amenable actions

� � � ∈ � H ′ Let Y = m ∈ ω � ( ∃ n ∈ Z ) p � � Q ξ “ m / ξ � n ” and note that � Y is finite. Let p ′ ≥ p be such that p ′ ( ξ + k ) = 1 ξ + k for each k ∈ Y \ S ∗ . Note that p ′ and q are compatible. Let q ′ extend both q and p ′ such that q ′ � Q “ � H ′ ξ � S ∗ = ˇ W ” and note that | W | ≤ � j ∈ S ∗ F ( j ) < | Z | . Let z ∈ Z \ W and note that, since q � Q “ � H ′ ξ � is a group”, it follows that ξ � z ∩ S ∗ = ∅ ”. q � Q “ � H ′ But since z ∈ Z it follows that if q ′ � Q “ m ∈ H ′ ξ z ” then ∈ � H ′ p � � Q ξ “ m / ξ � z ” and hence m ∈ Y ⊆ c ξ . In other words, q ′ � Q “ H ′ ξ z ⊆ c ξ ” and this contradicts the choice of H ξ using the fact that m G ( c ξ ) < 1. Juris Stepr¯ ans Amenable actions

To arrive at a contradiction construct, using the claim, a sequence, { ( p i , p ′ i , m i , m ′ i ) } i ∈ ω such that p i ∈ Q ξ and p ′ i ∈ Q η p i +1 ≤ p i ≤ q ↾ S ξ and p ′ i +1 ≤ p ′ i ≤ q ↾ S η p i � Q ξ “ m ′ i ∈ � H ′ ξ � m i ” p ′ i � Q η “ m i +1 ∈ � H ′ η � m ′ i ” all the m i and m ′ i are distinct. To carry out the induction it will be assumed as an additional induction hypothesis that � � � ∈ � H ′ X i = m ∈ ω � p i − 1 � � Q ξ “ m / ξ � m i ” is infinite � � p ′ X ′ � � ∈ � H ′ η � m ′ � i = m ∈ ω i − 1 � � Q η “ m / i ” is infinite. Juris Stepr¯ ans Amenable actions