Topological dynamics of stable groups Ludomir Newelski Instytut - - PowerPoint PPT Presentation

Topological dynamics of stable groups

Ludomir Newelski

Instytut Matematyczny Uniwersytet Wroc lawski

July 2017

Newelski Topological dynamics of stable groups

Set-up

G is a stable group in countable language L T = Th(G) G ∗ ≻ G is a monster model of T Can we do stable group theory without forking? Definition Let p, q ∈ S(G) p ∗ q = tp(a · b/M), where a | = p, b | = q and a⌣ | Gb (S(G), ∗) is a semi-group

Newelski Topological dynamics of stable groups

Set-up

G is a stable group in countable language L T = Th(G) G ∗ ≻ G is a monster model of T Can we do stable group theory without forking? Definition Let p, q ∈ S(G) p ∗ q = tp(a · b/M), where a | = p, b | = q and a⌣ | Gb (S(G), ∗) is a semi-group

Newelski Topological dynamics of stable groups

Set-up

G is a stable group in countable language L T = Th(G) G ∗ ≻ G is a monster model of T Can we do stable group theory without forking? Definition Let p, q ∈ S(G) p ∗ q = tp(a · b/M), where a | = p, b | = q and a⌣ | Gb (S(G), ∗) is a semi-group

Newelski Topological dynamics of stable groups

Set-up

G is a stable group in countable language L T = Th(G) G ∗ ≻ G is a monster model of T Can we do stable group theory without forking? Definition Let p, q ∈ S(G) p ∗ q = tp(a · b/M), where a | = p, b | = q and a⌣ | Gb (S(G), ∗) is a semi-group

Newelski Topological dynamics of stable groups

Set-up

G is a stable group in countable language L T = Th(G) G ∗ ≻ G is a monster model of T Can we do stable group theory without forking? Definition Let p, q ∈ S(G) p ∗ q = tp(a · b/M), where a | = p, b | = q and a⌣ | Gb (S(G), ∗) is a semi-group

Newelski Topological dynamics of stable groups

Set-up

G is a stable group in countable language L T = Th(G) G ∗ ≻ G is a monster model of T Can we do stable group theory without forking? Definition Let p, q ∈ S(G) p ∗ q = tp(a · b/M), where a | = p, b | = q and a⌣ | Gb (S(G), ∗) is a semi-group

Newelski Topological dynamics of stable groups

Topological dynamics

Definition (1) X is a G-flow if X is a compact topological space G acts on X by homeomorphisms (2) X is point-transitive if there is a dense G-orbit ⊆ X. (3) Y ⊆ X is a G-subflow of X if Y is closed and G-closed. Example Let X be a G-flow and p ∈ X. Then cl(Gp) is a subflow of X generated by p.

Newelski Topological dynamics of stable groups

Topological dynamics

Definition (1) X is a G-flow if X is a compact topological space G acts on X by homeomorphisms (2) X is point-transitive if there is a dense G-orbit ⊆ X. (3) Y ⊆ X is a G-subflow of X if Y is closed and G-closed. Example Let X be a G-flow and p ∈ X. Then cl(Gp) is a subflow of X generated by p.

Newelski Topological dynamics of stable groups

Topological dynamics

Definition (1) X is a G-flow if X is a compact topological space G acts on X by homeomorphisms (2) X is point-transitive if there is a dense G-orbit ⊆ X. (3) Y ⊆ X is a G-subflow of X if Y is closed and G-closed. Example Let X be a G-flow and p ∈ X. Then cl(Gp) is a subflow of X generated by p.

Newelski Topological dynamics of stable groups

Topological dynamics

Definition (1) X is a G-flow if X is a compact topological space G acts on X by homeomorphisms (2) X is point-transitive if there is a dense G-orbit ⊆ X. (3) Y ⊆ X is a G-subflow of X if Y is closed and G-closed. Example Let X be a G-flow and p ∈ X. Then cl(Gp) is a subflow of X generated by p.

Newelski Topological dynamics of stable groups

Topological dynamics

Definition (1) X is a G-flow if X is a compact topological space G acts on X by homeomorphisms (2) X is point-transitive if there is a dense G-orbit ⊆ X. (3) Y ⊆ X is a G-subflow of X if Y is closed and G-closed. Example Let X be a G-flow and p ∈ X. Then cl(Gp) is a subflow of X generated by p.

Newelski Topological dynamics of stable groups

Topological dynamics

Let X be a point-transitive G-flow. G ∋ g πg : X

≈

→ X, πg(x) = g · x, E(X) = cl({πg : g ∈ G}) ⊆ X X cl is the topological closure w.r. to pointwise convergence topology in X X E(X) is the Ellis (enveloping) semigroup of X E(X) is a point-transitive G-flow:

• 1. for f ∈ E(X) and g ∈ G, g ∗ f = πg ◦ f
• 2. {πg : g ∈ G} is a dense G-orbit.
• is continuous on E(X), in the first coordinate.

Newelski Topological dynamics of stable groups

Topological dynamics

Let X be a point-transitive G-flow. G ∋ g πg : X

≈

→ X, πg(x) = g · x, E(X) = cl({πg : g ∈ G}) ⊆ X X cl is the topological closure w.r. to pointwise convergence topology in X X E(X) is the Ellis (enveloping) semigroup of X E(X) is a point-transitive G-flow:

• 1. for f ∈ E(X) and g ∈ G, g ∗ f = πg ◦ f
• 2. {πg : g ∈ G} is a dense G-orbit.
• is continuous on E(X), in the first coordinate.

Newelski Topological dynamics of stable groups

Topological dynamics

Let X be a point-transitive G-flow. G ∋ g πg : X

≈

→ X, πg(x) = g · x, E(X) = cl({πg : g ∈ G}) ⊆ X X cl is the topological closure w.r. to pointwise convergence topology in X X E(X) is the Ellis (enveloping) semigroup of X E(X) is a point-transitive G-flow:

• 1. for f ∈ E(X) and g ∈ G, g ∗ f = πg ◦ f
• 2. {πg : g ∈ G} is a dense G-orbit.
• is continuous on E(X), in the first coordinate.

Newelski Topological dynamics of stable groups

Topological dynamics

Let X be a point-transitive G-flow. G ∋ g πg : X

≈

→ X, πg(x) = g · x, E(X) = cl({πg : g ∈ G}) ⊆ X X cl is the topological closure w.r. to pointwise convergence topology in X X E(X) is the Ellis (enveloping) semigroup of X E(X) is a point-transitive G-flow:

• 1. for f ∈ E(X) and g ∈ G, g ∗ f = πg ◦ f
• 2. {πg : g ∈ G} is a dense G-orbit.
• is continuous on E(X), in the first coordinate.

Newelski Topological dynamics of stable groups

Topological dynamics

Let X be a point-transitive G-flow. G ∋ g πg : X

≈

→ X, πg(x) = g · x, E(X) = cl({πg : g ∈ G}) ⊆ X X cl is the topological closure w.r. to pointwise convergence topology in X X E(X) is the Ellis (enveloping) semigroup of X E(X) is a point-transitive G-flow:

• 1. for f ∈ E(X) and g ∈ G, g ∗ f = πg ◦ f
• 2. {πg : g ∈ G} is a dense G-orbit.
• is continuous on E(X), in the first coordinate.

Newelski Topological dynamics of stable groups

Topological dynamics

Let X be a point-transitive G-flow. G ∋ g πg : X

≈

→ X, πg(x) = g · x, E(X) = cl({πg : g ∈ G}) ⊆ X X cl is the topological closure w.r. to pointwise convergence topology in X X E(X) is the Ellis (enveloping) semigroup of X E(X) is a point-transitive G-flow:

• 1. for f ∈ E(X) and g ∈ G, g ∗ f = πg ◦ f
• 2. {πg : g ∈ G} is a dense G-orbit.
• is continuous on E(X), in the first coordinate.

Newelski Topological dynamics of stable groups

Topological dynamics

Let X be a point-transitive G-flow. G ∋ g πg : X

≈

→ X, πg(x) = g · x, E(X) = cl({πg : g ∈ G}) ⊆ X X cl is the topological closure w.r. to pointwise convergence topology in X X E(X) is the Ellis (enveloping) semigroup of X E(X) is a point-transitive G-flow:

• 1. for f ∈ E(X) and g ∈ G, g ∗ f = πg ◦ f
• 2. {πg : g ∈ G} is a dense G-orbit.
• is continuous on E(X), in the first coordinate.

Newelski Topological dynamics of stable groups

Topological dynamics

Let X be a point-transitive G-flow. G ∋ g πg : X

≈

→ X, πg(x) = g · x, E(X) = cl({πg : g ∈ G}) ⊆ X X cl is the topological closure w.r. to pointwise convergence topology in X X E(X) is the Ellis (enveloping) semigroup of X E(X) is a point-transitive G-flow:

• 1. for f ∈ E(X) and g ∈ G, g ∗ f = πg ◦ f
• 2. {πg : g ∈ G} is a dense G-orbit.
• is continuous on E(X), in the first coordinate.

Newelski Topological dynamics of stable groups

Functional representation

Sometimes X ∼ = E(X) Let A ⊆ P(G) be a G-algebra of sets (i.e. closed under left translation in G). Then S(A) is a G-flow. For p ∈ S(A) we define dp : A → P(G) by: dp(U) = {g ∈ G : g−1U ∈ p} Definition A is d-closed if A is closed under dp for every p ∈ S(A). Example A = Def (G) is d-closed, because every p ∈ S(G) is definable.

Newelski Topological dynamics of stable groups

Functional representation

Sometimes X ∼ = E(X) Let A ⊆ P(G) be a G-algebra of sets (i.e. closed under left translation in G). Then S(A) is a G-flow. For p ∈ S(A) we define dp : A → P(G) by: dp(U) = {g ∈ G : g−1U ∈ p} Definition A is d-closed if A is closed under dp for every p ∈ S(A). Example A = Def (G) is d-closed, because every p ∈ S(G) is definable.

Newelski Topological dynamics of stable groups

Functional representation

Sometimes X ∼ = E(X) Let A ⊆ P(G) be a G-algebra of sets (i.e. closed under left translation in G). Then S(A) is a G-flow. For p ∈ S(A) we define dp : A → P(G) by: dp(U) = {g ∈ G : g−1U ∈ p} Definition A is d-closed if A is closed under dp for every p ∈ S(A). Example A = Def (G) is d-closed, because every p ∈ S(G) is definable.

Newelski Topological dynamics of stable groups

Functional representation

Sometimes X ∼ = E(X) Let A ⊆ P(G) be a G-algebra of sets (i.e. closed under left translation in G). Then S(A) is a G-flow. For p ∈ S(A) we define dp : A → P(G) by: dp(U) = {g ∈ G : g−1U ∈ p} Definition A is d-closed if A is closed under dp for every p ∈ S(A). Example A = Def (G) is d-closed, because every p ∈ S(G) is definable.

Newelski Topological dynamics of stable groups

Functional representation

Sometimes X ∼ = E(X) Let A ⊆ P(G) be a G-algebra of sets (i.e. closed under left translation in G). Then S(A) is a G-flow. For p ∈ S(A) we define dp : A → P(G) by: dp(U) = {g ∈ G : g−1U ∈ p} Definition A is d-closed if A is closed under dp for every p ∈ S(A). Example A = Def (G) is d-closed, because every p ∈ S(G) is definable.

Newelski Topological dynamics of stable groups

Functional representation

Sometimes X ∼ = E(X) Let A ⊆ P(G) be a G-algebra of sets (i.e. closed under left translation in G). Then S(A) is a G-flow. For p ∈ S(A) we define dp : A → P(G) by: dp(U) = {g ∈ G : g−1U ∈ p} Definition A is d-closed if A is closed under dp for every p ∈ S(A). Example A = Def (G) is d-closed, because every p ∈ S(G) is definable.

Newelski Topological dynamics of stable groups

Functional representation

Sometimes X ∼ = E(X) Let A ⊆ P(G) be a G-algebra of sets (i.e. closed under left translation in G). Then S(A) is a G-flow. For p ∈ S(A) we define dp : A → P(G) by: dp(U) = {g ∈ G : g−1U ∈ p} Definition A is d-closed if A is closed under dp for every p ∈ S(A). Example A = Def (G) is d-closed, because every p ∈ S(G) is definable.

Newelski Topological dynamics of stable groups

Functional representation

Sometimes X ∼ = E(X) Let A ⊆ P(G) be a G-algebra of sets (i.e. closed under left translation in G). Then S(A) is a G-flow. For p ∈ S(A) we define dp : A → P(G) by: dp(U) = {g ∈ G : g−1U ∈ p} Definition A is d-closed if A is closed under dp for every p ∈ S(A). Example A = Def (G) is d-closed, because every p ∈ S(G) is definable.

Newelski Topological dynamics of stable groups

Functional representation

Assume A is d-closed. For p ∈ S(A), dp ∈ End(A) := {G-endomorphisms of A}. Let d : S(A) → End(A) map p to dp. Then d is a bijection. d induces ∗ on S(A) so that d : (S(A), ∗)

∼ =

→ (End(A), ◦) Theorem 1 (E(S(A)), ◦) ∼ =1 (S(A), ∗) ∼ =2 (End(A), ◦) Proof

• 1. For p ∈ S(A) let lp(q) = p ∗ q.

Then lp ∈ E(S(A)) and p → lp gives ∼ =1.

• 2. This is d.

Newelski Topological dynamics of stable groups

Functional representation

Assume A is d-closed. For p ∈ S(A), dp ∈ End(A) := {G-endomorphisms of A}. Let d : S(A) → End(A) map p to dp. Then d is a bijection. d induces ∗ on S(A) so that d : (S(A), ∗)

∼ =

→ (End(A), ◦) Theorem 1 (E(S(A)), ◦) ∼ =1 (S(A), ∗) ∼ =2 (End(A), ◦) Proof

• 1. For p ∈ S(A) let lp(q) = p ∗ q.

Then lp ∈ E(S(A)) and p → lp gives ∼ =1.

• 2. This is d.

Newelski Topological dynamics of stable groups

Functional representation

Assume A is d-closed. For p ∈ S(A), dp ∈ End(A) := {G-endomorphisms of A}. Let d : S(A) → End(A) map p to dp. Then d is a bijection. d induces ∗ on S(A) so that d : (S(A), ∗)

∼ =

→ (End(A), ◦) Theorem 1 (E(S(A)), ◦) ∼ =1 (S(A), ∗) ∼ =2 (End(A), ◦) Proof

• 1. For p ∈ S(A) let lp(q) = p ∗ q.

Then lp ∈ E(S(A)) and p → lp gives ∼ =1.

• 2. This is d.

Newelski Topological dynamics of stable groups

Functional representation

Assume A is d-closed. For p ∈ S(A), dp ∈ End(A) := {G-endomorphisms of A}. Let d : S(A) → End(A) map p to dp. Then d is a bijection. d induces ∗ on S(A) so that d : (S(A), ∗)

∼ =

→ (End(A), ◦) Theorem 1 (E(S(A)), ◦) ∼ =1 (S(A), ∗) ∼ =2 (End(A), ◦) Proof

• 1. For p ∈ S(A) let lp(q) = p ∗ q.

Then lp ∈ E(S(A)) and p → lp gives ∼ =1.

• 2. This is d.

Newelski Topological dynamics of stable groups

Functional representation

Assume A is d-closed. For p ∈ S(A), dp ∈ End(A) := {G-endomorphisms of A}. Let d : S(A) → End(A) map p to dp. Then d is a bijection. d induces ∗ on S(A) so that d : (S(A), ∗)

∼ =

→ (End(A), ◦) Theorem 1 (E(S(A)), ◦) ∼ =1 (S(A), ∗) ∼ =2 (End(A), ◦) Proof

• 1. For p ∈ S(A) let lp(q) = p ∗ q.

Then lp ∈ E(S(A)) and p → lp gives ∼ =1.

• 2. This is d.

Newelski Topological dynamics of stable groups

Functional representation

Assume A is d-closed. For p ∈ S(A), dp ∈ End(A) := {G-endomorphisms of A}. Let d : S(A) → End(A) map p to dp. Then d is a bijection. d induces ∗ on S(A) so that d : (S(A), ∗)

∼ =

→ (End(A), ◦) Theorem 1 (E(S(A)), ◦) ∼ =1 (S(A), ∗) ∼ =2 (End(A), ◦) Proof

• 1. For p ∈ S(A) let lp(q) = p ∗ q.

Then lp ∈ E(S(A)) and p → lp gives ∼ =1.

• 2. This is d.

Newelski Topological dynamics of stable groups

Functional representation

Assume A is d-closed. For p ∈ S(A), dp ∈ End(A) := {G-endomorphisms of A}. Let d : S(A) → End(A) map p to dp. Then d is a bijection. d induces ∗ on S(A) so that d : (S(A), ∗)

∼ =

→ (End(A), ◦) Theorem 1 (E(S(A)), ◦) ∼ =1 (S(A), ∗) ∼ =2 (End(A), ◦) Proof

• 1. For p ∈ S(A) let lp(q) = p ∗ q.

Then lp ∈ E(S(A)) and p → lp gives ∼ =1.

• 2. This is d.

Newelski Topological dynamics of stable groups

Functional representation

Assume A is d-closed. For p ∈ S(A), dp ∈ End(A) := {G-endomorphisms of A}. Let d : S(A) → End(A) map p to dp. Then d is a bijection. d induces ∗ on S(A) so that d : (S(A), ∗)

∼ =

→ (End(A), ◦) Theorem 1 (E(S(A)), ◦) ∼ =1 (S(A), ∗) ∼ =2 (End(A), ◦) Proof

• 1. For p ∈ S(A) let lp(q) = p ∗ q.

Then lp ∈ E(S(A)) and p → lp gives ∼ =1.

• 2. This is d.

Newelski Topological dynamics of stable groups

Functional representation

Example If A = Def (G) then A is d-closed and ∗ on SG(M) = S(A) from Theorem 1 is just the free multiplication of G-types.

Newelski Topological dynamics of stable groups

Functional representation

Example If A = Def (G) then A is d-closed and ∗ on SG(M) = S(A) from Theorem 1 is just the free multiplication of G-types.

Newelski Topological dynamics of stable groups

(S(G), ∗) in the definable realm

Definition

• 1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsets
• f G is closed under left and right translation in G, and also under

taking inverse.

• 2. Let Inv = {∆ ⊆fin L : ∆ is invariant}.

Fact Inv is cofinal in [L]<ω. Let ∆ ∈ Inv. Notation Def∆(G) = {relatively ∆-definable subsets of G} S∆(G) = S(Def∆(G)), the space of complete ∆-types over G.

Newelski Topological dynamics of stable groups

(S(G), ∗) in the definable realm

Definition

• 1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsets
• f G is closed under left and right translation in G, and also under

taking inverse.

• 2. Let Inv = {∆ ⊆fin L : ∆ is invariant}.

Fact Inv is cofinal in [L]<ω. Let ∆ ∈ Inv. Notation Def∆(G) = {relatively ∆-definable subsets of G} S∆(G) = S(Def∆(G)), the space of complete ∆-types over G.

Newelski Topological dynamics of stable groups

(S(G), ∗) in the definable realm

Definition

• 1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsets
• f G is closed under left and right translation in G, and also under

taking inverse.

• 2. Let Inv = {∆ ⊆fin L : ∆ is invariant}.

Fact Inv is cofinal in [L]<ω. Let ∆ ∈ Inv. Notation Def∆(G) = {relatively ∆-definable subsets of G} S∆(G) = S(Def∆(G)), the space of complete ∆-types over G.

Newelski Topological dynamics of stable groups

(S(G), ∗) in the definable realm

Definition

• 1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsets
• f G is closed under left and right translation in G, and also under

taking inverse.

• 2. Let Inv = {∆ ⊆fin L : ∆ is invariant}.

Fact Inv is cofinal in [L]<ω. Let ∆ ∈ Inv. Notation Def∆(G) = {relatively ∆-definable subsets of G} S∆(G) = S(Def∆(G)), the space of complete ∆-types over G.

Newelski Topological dynamics of stable groups

(S(G), ∗) in the definable realm

Definition

• 1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsets
• f G is closed under left and right translation in G, and also under

taking inverse.

• 2. Let Inv = {∆ ⊆fin L : ∆ is invariant}.

Fact Inv is cofinal in [L]<ω. Let ∆ ∈ Inv. Notation Def∆(G) = {relatively ∆-definable subsets of G} S∆(G) = S(Def∆(G)), the space of complete ∆-types over G.

Newelski Topological dynamics of stable groups

(S(G), ∗) in the definable realm

Definition

• 1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsets
• f G is closed under left and right translation in G, and also under

taking inverse.

• 2. Let Inv = {∆ ⊆fin L : ∆ is invariant}.

Fact Inv is cofinal in [L]<ω. Let ∆ ∈ Inv. Notation Def∆(G) = {relatively ∆-definable subsets of G} S∆(G) = S(Def∆(G)), the space of complete ∆-types over G.

Newelski Topological dynamics of stable groups

(S(G), ∗) in the definable realm

Definition

• 1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsets
• f G is closed under left and right translation in G, and also under

taking inverse.

• 2. Let Inv = {∆ ⊆fin L : ∆ is invariant}.

Fact Inv is cofinal in [L]<ω. Let ∆ ∈ Inv. Notation Def∆(G) = {relatively ∆-definable subsets of G} S∆(G) = S(Def∆(G)), the space of complete ∆-types over G.

Newelski Topological dynamics of stable groups

(S(G), ∗) in the definable realm

Definition

• 1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsets
• f G is closed under left and right translation in G, and also under

taking inverse.

• 2. Let Inv = {∆ ⊆fin L : ∆ is invariant}.

Fact Inv is cofinal in [L]<ω. Let ∆ ∈ Inv. Notation Def∆(G) = {relatively ∆-definable subsets of G} S∆(G) = S(Def∆(G)), the space of complete ∆-types over G.

Newelski Topological dynamics of stable groups

(S(G), ∗) in the definable realm

Definition

• 1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsets
• f G is closed under left and right translation in G, and also under

taking inverse.

• 2. Let Inv = {∆ ⊆fin L : ∆ is invariant}.

Fact Inv is cofinal in [L]<ω. Let ∆ ∈ Inv. Notation Def∆(G) = {relatively ∆-definable subsets of G} S∆(G) = S(Def∆(G)), the space of complete ∆-types over G.

Newelski Topological dynamics of stable groups

(S(G), ∗) in the definable realm

1 DefG,∆(M) is a d-closed G-algebra of sets.

(this relies on the full definability lemma in local stability theory)

2 (SG,∆(M), ∗) ∼

= (E(SG,∆(M)), ◦) ∼ = (End(DefG,∆(M)), ◦) (this is by Theorem 1)

3

DefG(M) =

• ∆∈Inv

DefG,∆(M)

4 SG,∆(M), ∆ ∈ Inv is an inverse system of G- flows and

semi-groups (the connecting functions are restrictions)

5 SG(M) = invlim∆∈InvSG,∆(M)

(as G-flows and semigroups)

Newelski Topological dynamics of stable groups

(S(G), ∗) in the definable realm

1 DefG,∆(M) is a d-closed G-algebra of sets.

(this relies on the full definability lemma in local stability theory)

2 (SG,∆(M), ∗) ∼

= (E(SG,∆(M)), ◦) ∼ = (End(DefG,∆(M)), ◦) (this is by Theorem 1)

3

DefG(M) =

• ∆∈Inv

DefG,∆(M)

4 SG,∆(M), ∆ ∈ Inv is an inverse system of G- flows and

semi-groups (the connecting functions are restrictions)

5 SG(M) = invlim∆∈InvSG,∆(M)

(as G-flows and semigroups)

Newelski Topological dynamics of stable groups

(S(G), ∗) in the definable realm

1 DefG,∆(M) is a d-closed G-algebra of sets.

(this relies on the full definability lemma in local stability theory)

2 (SG,∆(M), ∗) ∼

= (E(SG,∆(M)), ◦) ∼ = (End(DefG,∆(M)), ◦) (this is by Theorem 1)

3

DefG(M) =

• ∆∈Inv

DefG,∆(M)

4 SG,∆(M), ∆ ∈ Inv is an inverse system of G- flows and

semi-groups (the connecting functions are restrictions)

5 SG(M) = invlim∆∈InvSG,∆(M)

(as G-flows and semigroups)

Newelski Topological dynamics of stable groups

(S(G), ∗) in the definable realm

1 DefG,∆(M) is a d-closed G-algebra of sets.

(this relies on the full definability lemma in local stability theory)

2 (SG,∆(M), ∗) ∼

= (E(SG,∆(M)), ◦) ∼ = (End(DefG,∆(M)), ◦) (this is by Theorem 1)

3

DefG(M) =

• ∆∈Inv

DefG,∆(M)

4 SG,∆(M), ∆ ∈ Inv is an inverse system of G- flows and

semi-groups (the connecting functions are restrictions)

5 SG(M) = invlim∆∈InvSG,∆(M)

(as G-flows and semigroups)

Newelski Topological dynamics of stable groups

(S(G), ∗) in the definable realm

1 DefG,∆(M) is a d-closed G-algebra of sets.

(this relies on the full definability lemma in local stability theory)

2 (SG,∆(M), ∗) ∼

= (E(SG,∆(M)), ◦) ∼ = (End(DefG,∆(M)), ◦) (this is by Theorem 1)

3

DefG(M) =

• ∆∈Inv

DefG,∆(M)

4 SG,∆(M), ∆ ∈ Inv is an inverse system of G- flows and

semi-groups (the connecting functions are restrictions)

5 SG(M) = invlim∆∈InvSG,∆(M)

(as G-flows and semigroups)

Newelski Topological dynamics of stable groups

(S(G), ∗) in the definable realm

1 DefG,∆(M) is a d-closed G-algebra of sets.

(this relies on the full definability lemma in local stability theory)

2 (SG,∆(M), ∗) ∼

= (E(SG,∆(M)), ◦) ∼ = (End(DefG,∆(M)), ◦) (this is by Theorem 1)

3

DefG(M) =

• ∆∈Inv

DefG,∆(M)

4 SG,∆(M), ∆ ∈ Inv is an inverse system of G- flows and

semi-groups (the connecting functions are restrictions)

5 SG(M) = invlim∆∈InvSG,∆(M)

(as G-flows and semigroups)

Newelski Topological dynamics of stable groups

(S(G), ∗) in the definable realm

1 DefG,∆(M) is a d-closed G-algebra of sets.

(this relies on the full definability lemma in local stability theory)

2 (SG,∆(M), ∗) ∼

= (E(SG,∆(M)), ◦) ∼ = (End(DefG,∆(M)), ◦) (this is by Theorem 1)

3

DefG(M) =

• ∆∈Inv

DefG,∆(M)

4 SG,∆(M), ∆ ∈ Inv is an inverse system of G- flows and

semi-groups (the connecting functions are restrictions)

5 SG(M) = invlim∆∈InvSG,∆(M)

(as G-flows and semigroups)

Newelski Topological dynamics of stable groups

(S(G), ∗) in the definable realm

1 DefG,∆(M) is a d-closed G-algebra of sets.

(this relies on the full definability lemma in local stability theory)

2 (SG,∆(M), ∗) ∼

= (E(SG,∆(M)), ◦) ∼ = (End(DefG,∆(M)), ◦) (this is by Theorem 1)

3

DefG(M) =

• ∆∈Inv

DefG,∆(M)

4 SG,∆(M), ∆ ∈ Inv is an inverse system of G- flows and

semi-groups (the connecting functions are restrictions)

5 SG(M) = invlim∆∈InvSG,∆(M)

(as G-flows and semigroups)

Newelski Topological dynamics of stable groups

Types as functions

S(G) ∋ p dp : Def (G) → Def (G) S∆(G) ∋ p dp : Def∆(G) → Def∆(G) dp Ker(dp), Im(dp) Ker(dp) = {U ∈ Def∆(G) : [U] ∩ cl(Gp) = ∅} Idea The larger the type p ∈ SG(M), p ∈ SG,∆(M) The smaller the flow cl(Gp). The larger the kernel Ker(dp). The smaller the image Im(dp). The larger the (local) Morley rank of p.

Newelski Topological dynamics of stable groups

Types as functions

S(G) ∋ p dp : Def (G) → Def (G) S∆(G) ∋ p dp : Def∆(G) → Def∆(G) dp Ker(dp), Im(dp) Ker(dp) = {U ∈ Def∆(G) : [U] ∩ cl(Gp) = ∅} Idea The larger the type p ∈ SG(M), p ∈ SG,∆(M) The smaller the flow cl(Gp). The larger the kernel Ker(dp). The smaller the image Im(dp). The larger the (local) Morley rank of p.

Newelski Topological dynamics of stable groups

Types as functions

S(G) ∋ p dp : Def (G) → Def (G) S∆(G) ∋ p dp : Def∆(G) → Def∆(G) dp Ker(dp), Im(dp) Ker(dp) = {U ∈ Def∆(G) : [U] ∩ cl(Gp) = ∅} Idea The larger the type p ∈ SG(M), p ∈ SG,∆(M) The smaller the flow cl(Gp). The larger the kernel Ker(dp). The smaller the image Im(dp). The larger the (local) Morley rank of p.

Newelski Topological dynamics of stable groups

Types as functions

S(G) ∋ p dp : Def (G) → Def (G) S∆(G) ∋ p dp : Def∆(G) → Def∆(G) dp Ker(dp), Im(dp) Ker(dp) = {U ∈ Def∆(G) : [U] ∩ cl(Gp) = ∅} Idea The larger the type p ∈ SG(M), p ∈ SG,∆(M) The smaller the flow cl(Gp). The larger the kernel Ker(dp). The smaller the image Im(dp). The larger the (local) Morley rank of p.

Newelski Topological dynamics of stable groups

Types as functions

S(G) ∋ p dp : Def (G) → Def (G) S∆(G) ∋ p dp : Def∆(G) → Def∆(G) dp Ker(dp), Im(dp) Ker(dp) = {U ∈ Def∆(G) : [U] ∩ cl(Gp) = ∅} Idea The larger the type p ∈ SG(M), p ∈ SG,∆(M) The smaller the flow cl(Gp). The larger the kernel Ker(dp). The smaller the image Im(dp). The larger the (local) Morley rank of p.

Newelski Topological dynamics of stable groups

Types as functions

S(G) ∋ p dp : Def (G) → Def (G) S∆(G) ∋ p dp : Def∆(G) → Def∆(G) dp Ker(dp), Im(dp) Ker(dp) = {U ∈ Def∆(G) : [U] ∩ cl(Gp) = ∅} Idea The larger the type p ∈ SG(M), p ∈ SG,∆(M) The smaller the flow cl(Gp). The larger the kernel Ker(dp). The smaller the image Im(dp). The larger the (local) Morley rank of p.

Newelski Topological dynamics of stable groups

Types as functions

S(G) ∋ p dp : Def (G) → Def (G) S∆(G) ∋ p dp : Def∆(G) → Def∆(G) dp Ker(dp), Im(dp) Ker(dp) = {U ∈ Def∆(G) : [U] ∩ cl(Gp) = ∅} Idea The larger the type p ∈ SG(M), p ∈ SG,∆(M) The smaller the flow cl(Gp). The larger the kernel Ker(dp). The smaller the image Im(dp). The larger the (local) Morley rank of p.

Newelski Topological dynamics of stable groups

Types as functions

S(G) ∋ p dp : Def (G) → Def (G) S∆(G) ∋ p dp : Def∆(G) → Def∆(G) dp Ker(dp), Im(dp) Ker(dp) = {U ∈ Def∆(G) : [U] ∩ cl(Gp) = ∅} Idea The larger the type p ∈ SG(M), p ∈ SG,∆(M) The smaller the flow cl(Gp). The larger the kernel Ker(dp). The smaller the image Im(dp). The larger the (local) Morley rank of p.

Newelski Topological dynamics of stable groups

Types as functions

Ker(dp), Im(dp): measures of the size of p. Let p ∈ S(G) (or p ∈ S∆(G)...) Let p∗n = p ∗ · · · ∗ p

• n

.So dp∗n = dp ◦ · · · ◦ dp

• n

. Let R(p) = RM∆(p) : ∆ ∈ Inv. Lemma

• 1. R(p∗n) grow (coordinatewise),Ker(dp∗n) grow and Im(dp∗n)

shrink with n = 1, 2, 3, . . . .

• 2. The growth/shrinking of these three sequences is strictly

correlated. There are similar connections between RM∆, Ker and Im in S∆(G). In particular, if p ∈ S∆(G), U ∈ Def∆(G) and RM∆(U) < RM∆(p), then U ∈ Ker(dp)

Newelski Topological dynamics of stable groups

Types as functions

Ker(dp), Im(dp): measures of the size of p. Let p ∈ S(G) (or p ∈ S∆(G)...) Let p∗n = p ∗ · · · ∗ p

• n

.So dp∗n = dp ◦ · · · ◦ dp

• n

. Let R(p) = RM∆(p) : ∆ ∈ Inv. Lemma

• 1. R(p∗n) grow (coordinatewise),Ker(dp∗n) grow and Im(dp∗n)

shrink with n = 1, 2, 3, . . . .

• 2. The growth/shrinking of these three sequences is strictly

correlated. There are similar connections between RM∆, Ker and Im in S∆(G). In particular, if p ∈ S∆(G), U ∈ Def∆(G) and RM∆(U) < RM∆(p), then U ∈ Ker(dp)

Newelski Topological dynamics of stable groups

Types as functions

Ker(dp), Im(dp): measures of the size of p. Let p ∈ S(G) (or p ∈ S∆(G)...) Let p∗n = p ∗ · · · ∗ p

• n

.So dp∗n = dp ◦ · · · ◦ dp

• n

. Let R(p) = RM∆(p) : ∆ ∈ Inv. Lemma

• 1. R(p∗n) grow (coordinatewise),Ker(dp∗n) grow and Im(dp∗n)

shrink with n = 1, 2, 3, . . . .

• 2. The growth/shrinking of these three sequences is strictly

correlated. There are similar connections between RM∆, Ker and Im in S∆(G). In particular, if p ∈ S∆(G), U ∈ Def∆(G) and RM∆(U) < RM∆(p), then U ∈ Ker(dp)

Newelski Topological dynamics of stable groups

Types as functions

Ker(dp), Im(dp): measures of the size of p. Let p ∈ S(G) (or p ∈ S∆(G)...) Let p∗n = p ∗ · · · ∗ p

• n

.So dp∗n = dp ◦ · · · ◦ dp

• n

. Let R(p) = RM∆(p) : ∆ ∈ Inv. Lemma

• 1. R(p∗n) grow (coordinatewise),Ker(dp∗n) grow and Im(dp∗n)

shrink with n = 1, 2, 3, . . . .

• 2. The growth/shrinking of these three sequences is strictly

correlated. There are similar connections between RM∆, Ker and Im in S∆(G). In particular, if p ∈ S∆(G), U ∈ Def∆(G) and RM∆(U) < RM∆(p), then U ∈ Ker(dp)

Newelski Topological dynamics of stable groups

Types as functions

Ker(dp), Im(dp): measures of the size of p. Let p ∈ S(G) (or p ∈ S∆(G)...) Let p∗n = p ∗ · · · ∗ p

• n

.So dp∗n = dp ◦ · · · ◦ dp

• n

. Let R(p) = RM∆(p) : ∆ ∈ Inv. Lemma

• 1. R(p∗n) grow (coordinatewise),Ker(dp∗n) grow and Im(dp∗n)

shrink with n = 1, 2, 3, . . . .

• 2. The growth/shrinking of these three sequences is strictly

correlated. There are similar connections between RM∆, Ker and Im in S∆(G). In particular, if p ∈ S∆(G), U ∈ Def∆(G) and RM∆(U) < RM∆(p), then U ∈ Ker(dp)

Newelski Topological dynamics of stable groups

Types as functions

Ker(dp), Im(dp): measures of the size of p. Let p ∈ S(G) (or p ∈ S∆(G)...) Let p∗n = p ∗ · · · ∗ p

• n

.So dp∗n = dp ◦ · · · ◦ dp

• n

. Let R(p) = RM∆(p) : ∆ ∈ Inv. Lemma

• 1. R(p∗n) grow (coordinatewise),Ker(dp∗n) grow and Im(dp∗n)

shrink with n = 1, 2, 3, . . . .

• 2. The growth/shrinking of these three sequences is strictly

correlated. There are similar connections between RM∆, Ker and Im in S∆(G). In particular, if p ∈ S∆(G), U ∈ Def∆(G) and RM∆(U) < RM∆(p), then U ∈ Ker(dp)

Newelski Topological dynamics of stable groups

Types as functions

Ker(dp), Im(dp): measures of the size of p. Let p ∈ S(G) (or p ∈ S∆(G)...) Let p∗n = p ∗ · · · ∗ p

• n

.So dp∗n = dp ◦ · · · ◦ dp

• n

. Let R(p) = RM∆(p) : ∆ ∈ Inv. Lemma

• 1. R(p∗n) grow (coordinatewise),Ker(dp∗n) grow and Im(dp∗n)

shrink with n = 1, 2, 3, . . . .

• 2. The growth/shrinking of these three sequences is strictly

correlated. There are similar connections between RM∆, Ker and Im in S∆(G). In particular, if p ∈ S∆(G), U ∈ Def∆(G) and RM∆(U) < RM∆(p), then U ∈ Ker(dp)

Newelski Topological dynamics of stable groups

Types as functions

Ker(dp), Im(dp): measures of the size of p. Let p ∈ S(G) (or p ∈ S∆(G)...) Let p∗n = p ∗ · · · ∗ p

• n

.So dp∗n = dp ◦ · · · ◦ dp

• n

. Let R(p) = RM∆(p) : ∆ ∈ Inv. Lemma

• 1. R(p∗n) grow (coordinatewise),Ker(dp∗n) grow and Im(dp∗n)

shrink with n = 1, 2, 3, . . . .

• 2. The growth/shrinking of these three sequences is strictly

correlated. There are similar connections between RM∆, Ker and Im in S∆(G). In particular, if p ∈ S∆(G), U ∈ Def∆(G) and RM∆(U) < RM∆(p), then U ∈ Ker(dp)

Newelski Topological dynamics of stable groups

Types as functions

Ker(dp), Im(dp): measures of the size of p. Let p ∈ S(G) (or p ∈ S∆(G)...) Let p∗n = p ∗ · · · ∗ p

• n

.So dp∗n = dp ◦ · · · ◦ dp

• n

. Let R(p) = RM∆(p) : ∆ ∈ Inv. Lemma

• 1. R(p∗n) grow (coordinatewise),Ker(dp∗n) grow and Im(dp∗n)

shrink with n = 1, 2, 3, . . . .

• 2. The growth/shrinking of these three sequences is strictly

correlated. There are similar connections between RM∆, Ker and Im in S∆(G). In particular, if p ∈ S∆(G), U ∈ Def∆(G) and RM∆(U) < RM∆(p), then U ∈ Ker(dp)

Newelski Topological dynamics of stable groups

Subalgebras of Def∆(G)

Assume A ⊆ Def∆(G) is a G-subalgebra. A is scattered, has finite CB-rank, MR∆-rank. A is atomic. For g ∈ G let Ug ∈ A be the atom containing g. It exists: there is some atom U ∈ A, then U = Uh for any h ∈ U. then 1 ∈ h−1Uh = U1 g ∈ gU1 = Ug U1 < G Ug = gU1 is the left coset of U1 containing g. Atoms almost determine A

Newelski Topological dynamics of stable groups

Subalgebras of Def∆(G)

Assume A ⊆ Def∆(G) is a G-subalgebra. A is scattered, has finite CB-rank, MR∆-rank. A is atomic. For g ∈ G let Ug ∈ A be the atom containing g. It exists: there is some atom U ∈ A, then U = Uh for any h ∈ U. then 1 ∈ h−1Uh = U1 g ∈ gU1 = Ug U1 < G Ug = gU1 is the left coset of U1 containing g. Atoms almost determine A

Newelski Topological dynamics of stable groups

Subalgebras of Def∆(G)

Assume A ⊆ Def∆(G) is a G-subalgebra. A is scattered, has finite CB-rank, MR∆-rank. A is atomic. For g ∈ G let Ug ∈ A be the atom containing g. It exists: there is some atom U ∈ A, then U = Uh for any h ∈ U. then 1 ∈ h−1Uh = U1 g ∈ gU1 = Ug U1 < G Ug = gU1 is the left coset of U1 containing g. Atoms almost determine A

Newelski Topological dynamics of stable groups

Subalgebras of Def∆(G)

Assume A ⊆ Def∆(G) is a G-subalgebra. A is scattered, has finite CB-rank, MR∆-rank. A is atomic. For g ∈ G let Ug ∈ A be the atom containing g. It exists: there is some atom U ∈ A, then U = Uh for any h ∈ U. then 1 ∈ h−1Uh = U1 g ∈ gU1 = Ug U1 < G Ug = gU1 is the left coset of U1 containing g. Atoms almost determine A

Newelski Topological dynamics of stable groups

Subalgebras of Def∆(G)

Assume A ⊆ Def∆(G) is a G-subalgebra. A is scattered, has finite CB-rank, MR∆-rank. A is atomic. For g ∈ G let Ug ∈ A be the atom containing g. It exists: there is some atom U ∈ A, then U = Uh for any h ∈ U. then 1 ∈ h−1Uh = U1 g ∈ gU1 = Ug U1 < G Ug = gU1 is the left coset of U1 containing g. Atoms almost determine A

Newelski Topological dynamics of stable groups

Subalgebras of Def∆(G)

Assume A ⊆ Def∆(G) is a G-subalgebra. A is scattered, has finite CB-rank, MR∆-rank. A is atomic. For g ∈ G let Ug ∈ A be the atom containing g. It exists: there is some atom U ∈ A, then U = Uh for any h ∈ U. then 1 ∈ h−1Uh = U1 g ∈ gU1 = Ug U1 < G Ug = gU1 is the left coset of U1 containing g. Atoms almost determine A

Newelski Topological dynamics of stable groups

Subalgebras of Def∆(G)

Assume A ⊆ Def∆(G) is a G-subalgebra. A is scattered, has finite CB-rank, MR∆-rank. A is atomic. For g ∈ G let Ug ∈ A be the atom containing g. It exists: there is some atom U ∈ A, then U = Uh for any h ∈ U. then 1 ∈ h−1Uh = U1 g ∈ gU1 = Ug U1 < G Ug = gU1 is the left coset of U1 containing g. Atoms almost determine A

Newelski Topological dynamics of stable groups

Subalgebras of Def∆(G)

Assume A ⊆ Def∆(G) is a G-subalgebra. A is scattered, has finite CB-rank, MR∆-rank. A is atomic. For g ∈ G let Ug ∈ A be the atom containing g. It exists: there is some atom U ∈ A, then U = Uh for any h ∈ U. then 1 ∈ h−1Uh = U1 g ∈ gU1 = Ug U1 < G Ug = gU1 is the left coset of U1 containing g. Atoms almost determine A

Newelski Topological dynamics of stable groups

Subalgebras of Def∆(G)

Assume A ⊆ Def∆(G) is a G-subalgebra. A is scattered, has finite CB-rank, MR∆-rank. A is atomic. For g ∈ G let Ug ∈ A be the atom containing g. It exists: there is some atom U ∈ A, then U = Uh for any h ∈ U. then 1 ∈ h−1Uh = U1 g ∈ gU1 = Ug U1 < G Ug = gU1 is the left coset of U1 containing g. Atoms almost determine A

Newelski Topological dynamics of stable groups

Subalgebras of Def∆(G)

Assume A ⊆ Def∆(G) is a G-subalgebra. A is scattered, has finite CB-rank, MR∆-rank. A is atomic. For g ∈ G let Ug ∈ A be the atom containing g. It exists: there is some atom U ∈ A, then U = Uh for any h ∈ U. then 1 ∈ h−1Uh = U1 g ∈ gU1 = Ug U1 < G Ug = gU1 is the left coset of U1 containing g. Atoms almost determine A

Newelski Topological dynamics of stable groups

A := Im(dp) explained

Let p ∈ S∆(G) and A = Im(dp). What is U1? For V ∈ Def ∆(G): 1 ∈ dp(V ) ⇐ ⇒ V ∈ p Choose V ∈ p with CB(V ) = CB(p) and Mlt(V ) = Mlt(p). Here CB and Mlt is meant in Def∆(G). Then U1 = dp(V ) Ug = gU1 for all g ∈ G In fact, U1 = Stab(p) = {g ∈ G : gp = p}.

Newelski Topological dynamics of stable groups

A := Im(dp) explained

Let p ∈ S∆(G) and A = Im(dp). What is U1? For V ∈ Def ∆(G): 1 ∈ dp(V ) ⇐ ⇒ V ∈ p Choose V ∈ p with CB(V ) = CB(p) and Mlt(V ) = Mlt(p). Here CB and Mlt is meant in Def∆(G). Then U1 = dp(V ) Ug = gU1 for all g ∈ G In fact, U1 = Stab(p) = {g ∈ G : gp = p}.

Newelski Topological dynamics of stable groups

A := Im(dp) explained

Let p ∈ S∆(G) and A = Im(dp). What is U1? For V ∈ Def ∆(G): 1 ∈ dp(V ) ⇐ ⇒ V ∈ p Choose V ∈ p with CB(V ) = CB(p) and Mlt(V ) = Mlt(p). Here CB and Mlt is meant in Def∆(G). Then U1 = dp(V ) Ug = gU1 for all g ∈ G In fact, U1 = Stab(p) = {g ∈ G : gp = p}.

Newelski Topological dynamics of stable groups

A := Im(dp) explained

Let p ∈ S∆(G) and A = Im(dp). What is U1? For V ∈ Def ∆(G): 1 ∈ dp(V ) ⇐ ⇒ V ∈ p Choose V ∈ p with CB(V ) = CB(p) and Mlt(V ) = Mlt(p). Here CB and Mlt is meant in Def∆(G). Then U1 = dp(V ) Ug = gU1 for all g ∈ G In fact, U1 = Stab(p) = {g ∈ G : gp = p}.

Newelski Topological dynamics of stable groups

A := Im(dp) explained

Let p ∈ S∆(G) and A = Im(dp). What is U1? For V ∈ Def ∆(G): 1 ∈ dp(V ) ⇐ ⇒ V ∈ p Choose V ∈ p with CB(V ) = CB(p) and Mlt(V ) = Mlt(p). Here CB and Mlt is meant in Def∆(G). Then U1 = dp(V ) Ug = gU1 for all g ∈ G In fact, U1 = Stab(p) = {g ∈ G : gp = p}.

Newelski Topological dynamics of stable groups

A := Im(dp) explained

Let p ∈ S∆(G) and A = Im(dp). What is U1? For V ∈ Def ∆(G): 1 ∈ dp(V ) ⇐ ⇒ V ∈ p Choose V ∈ p with CB(V ) = CB(p) and Mlt(V ) = Mlt(p). Here CB and Mlt is meant in Def∆(G). Then U1 = dp(V ) Ug = gU1 for all g ∈ G In fact, U1 = Stab(p) = {g ∈ G : gp = p}.

Newelski Topological dynamics of stable groups

A := Im(dp) explained

Let p ∈ S∆(G) and A = Im(dp). What is U1? For V ∈ Def ∆(G): 1 ∈ dp(V ) ⇐ ⇒ V ∈ p Choose V ∈ p with CB(V ) = CB(p) and Mlt(V ) = Mlt(p). Here CB and Mlt is meant in Def∆(G). Then U1 = dp(V ) Ug = gU1 for all g ∈ G In fact, U1 = Stab(p) = {g ∈ G : gp = p}.

Newelski Topological dynamics of stable groups

A := Im(dp) explained

Let p ∈ S∆(G) and A = Im(dp). What is U1? For V ∈ Def ∆(G): 1 ∈ dp(V ) ⇐ ⇒ V ∈ p Choose V ∈ p with CB(V ) = CB(p) and Mlt(V ) = Mlt(p). Here CB and Mlt is meant in Def∆(G). Then U1 = dp(V ) Ug = gU1 for all g ∈ G In fact, U1 = Stab(p) = {g ∈ G : gp = p}.

Newelski Topological dynamics of stable groups

Test case: the 2-step theorem

Assume p ∈ S(G). Recall that G ∗ ≻ G is a monster model. p(G ∗) generates a subgroup p(G ∗) < G ∗, invariant under Aut(G ∗/G). Let p be the minimal type-definable subgroup of G ∗ containing p(G ∗). Let Cl∗(p) = cl({pn : n ∈ N+}), where pn = p ∗ · · · ∗ p

• n

So pn(G ∗) ⊆ p. Theorem 2 The generic types of p are precisely the types in Cl∗(p) with maximal ranks RM∆, ∆ ⊆ L finite, invariant.

Newelski Topological dynamics of stable groups

Test case: the 2-step theorem

Assume p ∈ S(G). Recall that G ∗ ≻ G is a monster model. p(G ∗) generates a subgroup p(G ∗) < G ∗, invariant under Aut(G ∗/G). Let p be the minimal type-definable subgroup of G ∗ containing p(G ∗). Let Cl∗(p) = cl({pn : n ∈ N+}), where pn = p ∗ · · · ∗ p

• n

So pn(G ∗) ⊆ p. Theorem 2 The generic types of p are precisely the types in Cl∗(p) with maximal ranks RM∆, ∆ ⊆ L finite, invariant.

Newelski Topological dynamics of stable groups

Test case: the 2-step theorem

Assume p ∈ S(G). Recall that G ∗ ≻ G is a monster model. p(G ∗) generates a subgroup p(G ∗) < G ∗, invariant under Aut(G ∗/G). Let p be the minimal type-definable subgroup of G ∗ containing p(G ∗). Let Cl∗(p) = cl({pn : n ∈ N+}), where pn = p ∗ · · · ∗ p

• n

So pn(G ∗) ⊆ p. Theorem 2 The generic types of p are precisely the types in Cl∗(p) with maximal ranks RM∆, ∆ ⊆ L finite, invariant.

Newelski Topological dynamics of stable groups

Test case: the 2-step theorem

Assume p ∈ S(G). Recall that G ∗ ≻ G is a monster model. p(G ∗) generates a subgroup p(G ∗) < G ∗, invariant under Aut(G ∗/G). Let p be the minimal type-definable subgroup of G ∗ containing p(G ∗). Let Cl∗(p) = cl({pn : n ∈ N+}), where pn = p ∗ · · · ∗ p

• n

So pn(G ∗) ⊆ p. Theorem 2 The generic types of p are precisely the types in Cl∗(p) with maximal ranks RM∆, ∆ ⊆ L finite, invariant.

Newelski Topological dynamics of stable groups

Test case: the 2-step theorem

Assume p ∈ S(G). Recall that G ∗ ≻ G is a monster model. p(G ∗) generates a subgroup p(G ∗) < G ∗, invariant under Aut(G ∗/G). Let p be the minimal type-definable subgroup of G ∗ containing p(G ∗). Let Cl∗(p) = cl({pn : n ∈ N+}), where pn = p ∗ · · · ∗ p

• n

So pn(G ∗) ⊆ p. Theorem 2 The generic types of p are precisely the types in Cl∗(p) with maximal ranks RM∆, ∆ ⊆ L finite, invariant.

Newelski Topological dynamics of stable groups

Test case: the 2-step theorem

Assume p ∈ S(G). Recall that G ∗ ≻ G is a monster model. p(G ∗) generates a subgroup p(G ∗) < G ∗, invariant under Aut(G ∗/G). Let p be the minimal type-definable subgroup of G ∗ containing p(G ∗). Let Cl∗(p) = cl({pn : n ∈ N+}), where pn = p ∗ · · · ∗ p

• n

So pn(G ∗) ⊆ p. Theorem 2 The generic types of p are precisely the types in Cl∗(p) with maximal ranks RM∆, ∆ ⊆ L finite, invariant.

Newelski Topological dynamics of stable groups

The 2-step theorem

RM∆(p0 ∗ p1) ≥ RM∆(pi) RM∆(pn), n < ω, is non-decreasing. If q = limi pni, then RM∆(q) ≥ RM∆(pni). If q ∈ Cl∗(p) is a generic type of p, then q is an accumulation point of the set {pn : n > 0}. Is the converse true? Not necessarily. The 2-step theorem Assume q is an accumulation point of {pn : n > 0} and r is an accumulation point of {qn : n > 0}. Then r is a generic type of p.

Newelski Topological dynamics of stable groups

The 2-step theorem

RM∆(p0 ∗ p1) ≥ RM∆(pi) RM∆(pn), n < ω, is non-decreasing. If q = limi pni, then RM∆(q) ≥ RM∆(pni). If q ∈ Cl∗(p) is a generic type of p, then q is an accumulation point of the set {pn : n > 0}. Is the converse true? Not necessarily. The 2-step theorem Assume q is an accumulation point of {pn : n > 0} and r is an accumulation point of {qn : n > 0}. Then r is a generic type of p.

Newelski Topological dynamics of stable groups

The 2-step theorem

RM∆(p0 ∗ p1) ≥ RM∆(pi) RM∆(pn), n < ω, is non-decreasing. If q = limi pni, then RM∆(q) ≥ RM∆(pni). If q ∈ Cl∗(p) is a generic type of p, then q is an accumulation point of the set {pn : n > 0}. Is the converse true? Not necessarily. The 2-step theorem Assume q is an accumulation point of {pn : n > 0} and r is an accumulation point of {qn : n > 0}. Then r is a generic type of p.

Newelski Topological dynamics of stable groups

The 2-step theorem

RM∆(p0 ∗ p1) ≥ RM∆(pi) RM∆(pn), n < ω, is non-decreasing. If q = limi pni, then RM∆(q) ≥ RM∆(pni). If q ∈ Cl∗(p) is a generic type of p, then q is an accumulation point of the set {pn : n > 0}. Is the converse true? Not necessarily. The 2-step theorem Assume q is an accumulation point of {pn : n > 0} and r is an accumulation point of {qn : n > 0}. Then r is a generic type of p.

Newelski Topological dynamics of stable groups

The 2-step theorem

RM∆(p0 ∗ p1) ≥ RM∆(pi) RM∆(pn), n < ω, is non-decreasing. If q = limi pni, then RM∆(q) ≥ RM∆(pni). If q ∈ Cl∗(p) is a generic type of p, then q is an accumulation point of the set {pn : n > 0}. Is the converse true? Not necessarily. The 2-step theorem Assume q is an accumulation point of {pn : n > 0} and r is an accumulation point of {qn : n > 0}. Then r is a generic type of p.

Newelski Topological dynamics of stable groups

The 2-step theorem

RM∆(p0 ∗ p1) ≥ RM∆(pi) RM∆(pn), n < ω, is non-decreasing. If q = limi pni, then RM∆(q) ≥ RM∆(pni). If q ∈ Cl∗(p) is a generic type of p, then q is an accumulation point of the set {pn : n > 0}. Is the converse true? Not necessarily. The 2-step theorem Assume q is an accumulation point of {pn : n > 0} and r is an accumulation point of {qn : n > 0}. Then r is a generic type of p.

Newelski Topological dynamics of stable groups

An example

Let p ∈ S(G) or S∆(G) and q = p−1 ∗ p. Then the group q is connected.

• Proof. Wlog p ∈ S∆(G). So q ∈ S∆(G).

Im(dq1) ⊇ Im(dq2) ⊇ Im(dq3) ⊇ . . . There is n s.t. Im(dqn) = Im(dqn+k) for all k. Let A = Im(dqn). So dqn+1 = dp−1 ◦ dp ◦ dqn. We shall prove that dq|A = idA dp|A is 1-1 dp : A

∼ =

→ A′ ⊆ Def∆(G)

Newelski Topological dynamics of stable groups

An example

Let p ∈ S(G) or S∆(G) and q = p−1 ∗ p. Then the group q is connected.

• Proof. Wlog p ∈ S∆(G). So q ∈ S∆(G).

Im(dq1) ⊇ Im(dq2) ⊇ Im(dq3) ⊇ . . . There is n s.t. Im(dqn) = Im(dqn+k) for all k. Let A = Im(dqn). So dqn+1 = dp−1 ◦ dp ◦ dqn. We shall prove that dq|A = idA dp|A is 1-1 dp : A

∼ =

→ A′ ⊆ Def∆(G)

Newelski Topological dynamics of stable groups

An example

Let p ∈ S(G) or S∆(G) and q = p−1 ∗ p. Then the group q is connected.

• Proof. Wlog p ∈ S∆(G). So q ∈ S∆(G).

Im(dq1) ⊇ Im(dq2) ⊇ Im(dq3) ⊇ . . . There is n s.t. Im(dqn) = Im(dqn+k) for all k. Let A = Im(dqn). So dqn+1 = dp−1 ◦ dp ◦ dqn. We shall prove that dq|A = idA dp|A is 1-1 dp : A

∼ =

→ A′ ⊆ Def∆(G)

Newelski Topological dynamics of stable groups

An example

Let p ∈ S(G) or S∆(G) and q = p−1 ∗ p. Then the group q is connected.

• Proof. Wlog p ∈ S∆(G). So q ∈ S∆(G).

Im(dq1) ⊇ Im(dq2) ⊇ Im(dq3) ⊇ . . . There is n s.t. Im(dqn) = Im(dqn+k) for all k. Let A = Im(dqn). So dqn+1 = dp−1 ◦ dp ◦ dqn. We shall prove that dq|A = idA dp|A is 1-1 dp : A

∼ =

→ A′ ⊆ Def∆(G)

Newelski Topological dynamics of stable groups

An example

Let p ∈ S(G) or S∆(G) and q = p−1 ∗ p. Then the group q is connected.

• Proof. Wlog p ∈ S∆(G). So q ∈ S∆(G).

Im(dq1) ⊇ Im(dq2) ⊇ Im(dq3) ⊇ . . . There is n s.t. Im(dqn) = Im(dqn+k) for all k. Let A = Im(dqn). So dqn+1 = dp−1 ◦ dp ◦ dqn. We shall prove that dq|A = idA dp|A is 1-1 dp : A

∼ =

→ A′ ⊆ Def∆(G)

Newelski Topological dynamics of stable groups

An example

Let p ∈ S(G) or S∆(G) and q = p−1 ∗ p. Then the group q is connected.

• Proof. Wlog p ∈ S∆(G). So q ∈ S∆(G).

Im(dq1) ⊇ Im(dq2) ⊇ Im(dq3) ⊇ . . . There is n s.t. Im(dqn) = Im(dqn+k) for all k. Let A = Im(dqn). So dqn+1 = dp−1 ◦ dp ◦ dqn. We shall prove that dq|A = idA dp|A is 1-1 dp : A

∼ =

→ A′ ⊆ Def∆(G)

Newelski Topological dynamics of stable groups

An example

Let p ∈ S(G) or S∆(G) and q = p−1 ∗ p. Then the group q is connected.

• Proof. Wlog p ∈ S∆(G). So q ∈ S∆(G).

Im(dq1) ⊇ Im(dq2) ⊇ Im(dq3) ⊇ . . . There is n s.t. Im(dqn) = Im(dqn+k) for all k. Let A = Im(dqn). So dqn+1 = dp−1 ◦ dp ◦ dqn. We shall prove that dq|A = idA dp|A is 1-1 dp : A

∼ =

→ A′ ⊆ Def∆(G)

Newelski Topological dynamics of stable groups

The proof continued

dp(U1) = ∅, so Us = sU1 ∈ p for some s ∈ G. U′

1 = sU1s−1 = Us 1 = dp(Us)

dp(Uh) = dp(hs−1Us) = hs−1dp(Us) = hs−1U′

1 = U′ hs−1

dp−1 : A′ ∼

=

→ A (as Im(dqn+1) = Im(dqn)). dp−1(U′

1) = 0, so U′ t = tU′ 1 ∈ p−1 for some t ∈ G.

Can take t = s−1: sU1 ∈ p ⇒ (sU1)−1 = U1s−1 ∈ p−1 U1s−1 = s−1sU1s−1 = s−1U′

1

dq(U1) = (dp−1 ◦ dp)(U1) = dp−1(U′

s−1) = Us−1t−1 = U1

dq(Uh) = dq(hU1) = hdq(U1) = hU1 = Uh So dq is identity on A.

Newelski Topological dynamics of stable groups

The proof continued

dp(U1) = ∅, so Us = sU1 ∈ p for some s ∈ G. U′

1 = sU1s−1 = Us 1 = dp(Us)

dp(Uh) = dp(hs−1Us) = hs−1dp(Us) = hs−1U′

1 = U′ hs−1

dp−1 : A′ ∼

=

→ A (as Im(dqn+1) = Im(dqn)). dp−1(U′

1) = 0, so U′ t = tU′ 1 ∈ p−1 for some t ∈ G.

Can take t = s−1: sU1 ∈ p ⇒ (sU1)−1 = U1s−1 ∈ p−1 U1s−1 = s−1sU1s−1 = s−1U′

1

dq(U1) = (dp−1 ◦ dp)(U1) = dp−1(U′

s−1) = Us−1t−1 = U1

dq(Uh) = dq(hU1) = hdq(U1) = hU1 = Uh So dq is identity on A.

Newelski Topological dynamics of stable groups

The proof continued

dp(U1) = ∅, so Us = sU1 ∈ p for some s ∈ G. U′

1 = sU1s−1 = Us 1 = dp(Us)

dp(Uh) = dp(hs−1Us) = hs−1dp(Us) = hs−1U′

1 = U′ hs−1

dp−1 : A′ ∼

=

→ A (as Im(dqn+1) = Im(dqn)). dp−1(U′

1) = 0, so U′ t = tU′ 1 ∈ p−1 for some t ∈ G.

Can take t = s−1: sU1 ∈ p ⇒ (sU1)−1 = U1s−1 ∈ p−1 U1s−1 = s−1sU1s−1 = s−1U′

1

dq(U1) = (dp−1 ◦ dp)(U1) = dp−1(U′

s−1) = Us−1t−1 = U1

dq(Uh) = dq(hU1) = hdq(U1) = hU1 = Uh So dq is identity on A.

Newelski Topological dynamics of stable groups

The proof continued

dp(U1) = ∅, so Us = sU1 ∈ p for some s ∈ G. U′

1 = sU1s−1 = Us 1 = dp(Us)

dp(Uh) = dp(hs−1Us) = hs−1dp(Us) = hs−1U′

1 = U′ hs−1

dp−1 : A′ ∼

=

→ A (as Im(dqn+1) = Im(dqn)). dp−1(U′

1) = 0, so U′ t = tU′ 1 ∈ p−1 for some t ∈ G.

Can take t = s−1: sU1 ∈ p ⇒ (sU1)−1 = U1s−1 ∈ p−1 U1s−1 = s−1sU1s−1 = s−1U′

1

dq(U1) = (dp−1 ◦ dp)(U1) = dp−1(U′

s−1) = Us−1t−1 = U1

dq(Uh) = dq(hU1) = hdq(U1) = hU1 = Uh So dq is identity on A.

Newelski Topological dynamics of stable groups

The proof continued

dp(U1) = ∅, so Us = sU1 ∈ p for some s ∈ G. U′

1 = sU1s−1 = Us 1 = dp(Us)

dp(Uh) = dp(hs−1Us) = hs−1dp(Us) = hs−1U′

1 = U′ hs−1

dp−1 : A′ ∼

=

→ A (as Im(dqn+1) = Im(dqn)). dp−1(U′

1) = 0, so U′ t = tU′ 1 ∈ p−1 for some t ∈ G.

Can take t = s−1: sU1 ∈ p ⇒ (sU1)−1 = U1s−1 ∈ p−1 U1s−1 = s−1sU1s−1 = s−1U′

1

dq(U1) = (dp−1 ◦ dp)(U1) = dp−1(U′

s−1) = Us−1t−1 = U1

dq(Uh) = dq(hU1) = hdq(U1) = hU1 = Uh So dq is identity on A.

Newelski Topological dynamics of stable groups

The proof continued

dp(U1) = ∅, so Us = sU1 ∈ p for some s ∈ G. U′

1 = sU1s−1 = Us 1 = dp(Us)

dp(Uh) = dp(hs−1Us) = hs−1dp(Us) = hs−1U′

1 = U′ hs−1

dp−1 : A′ ∼

=

→ A (as Im(dqn+1) = Im(dqn)). dp−1(U′

1) = 0, so U′ t = tU′ 1 ∈ p−1 for some t ∈ G.

Can take t = s−1: sU1 ∈ p ⇒ (sU1)−1 = U1s−1 ∈ p−1 U1s−1 = s−1sU1s−1 = s−1U′

1

dq(U1) = (dp−1 ◦ dp)(U1) = dp−1(U′

s−1) = Us−1t−1 = U1

dq(Uh) = dq(hU1) = hdq(U1) = hU1 = Uh So dq is identity on A.

Newelski Topological dynamics of stable groups

The proof continued

dp(U1) = ∅, so Us = sU1 ∈ p for some s ∈ G. U′

1 = sU1s−1 = Us 1 = dp(Us)

dp(Uh) = dp(hs−1Us) = hs−1dp(Us) = hs−1U′

1 = U′ hs−1

dp−1 : A′ ∼

=

→ A (as Im(dqn+1) = Im(dqn)). dp−1(U′

1) = 0, so U′ t = tU′ 1 ∈ p−1 for some t ∈ G.

Can take t = s−1: sU1 ∈ p ⇒ (sU1)−1 = U1s−1 ∈ p−1 U1s−1 = s−1sU1s−1 = s−1U′

1

dq(U1) = (dp−1 ◦ dp)(U1) = dp−1(U′

s−1) = Us−1t−1 = U1

dq(Uh) = dq(hU1) = hdq(U1) = hU1 = Uh So dq is identity on A.

Newelski Topological dynamics of stable groups

The proof continued

dp(U1) = ∅, so Us = sU1 ∈ p for some s ∈ G. U′

1 = sU1s−1 = Us 1 = dp(Us)

dp(Uh) = dp(hs−1Us) = hs−1dp(Us) = hs−1U′

1 = U′ hs−1

dp−1 : A′ ∼

=

→ A (as Im(dqn+1) = Im(dqn)). dp−1(U′

1) = 0, so U′ t = tU′ 1 ∈ p−1 for some t ∈ G.

Can take t = s−1: sU1 ∈ p ⇒ (sU1)−1 = U1s−1 ∈ p−1 U1s−1 = s−1sU1s−1 = s−1U′

1

dq(U1) = (dp−1 ◦ dp)(U1) = dp−1(U′

s−1) = Us−1t−1 = U1

dq(Uh) = dq(hU1) = hdq(U1) = hU1 = Uh So dq is identity on A.

Newelski Topological dynamics of stable groups

The proof continued

dp(U1) = ∅, so Us = sU1 ∈ p for some s ∈ G. U′

1 = sU1s−1 = Us 1 = dp(Us)

dp(Uh) = dp(hs−1Us) = hs−1dp(Us) = hs−1U′

1 = U′ hs−1

dp−1 : A′ ∼

=

→ A (as Im(dqn+1) = Im(dqn)). dp−1(U′

1) = 0, so U′ t = tU′ 1 ∈ p−1 for some t ∈ G.

Can take t = s−1: sU1 ∈ p ⇒ (sU1)−1 = U1s−1 ∈ p−1 U1s−1 = s−1sU1s−1 = s−1U′

1

dq(U1) = (dp−1 ◦ dp)(U1) = dp−1(U′

s−1) = Us−1t−1 = U1

dq(Uh) = dq(hU1) = hdq(U1) = hU1 = Uh So dq is identity on A.

Newelski Topological dynamics of stable groups

The proof continued

dp(U1) = ∅, so Us = sU1 ∈ p for some s ∈ G. U′

1 = sU1s−1 = Us 1 = dp(Us)

dp(Uh) = dp(hs−1Us) = hs−1dp(Us) = hs−1U′

1 = U′ hs−1

dp−1 : A′ ∼

=

→ A (as Im(dqn+1) = Im(dqn)). dp−1(U′

1) = 0, so U′ t = tU′ 1 ∈ p−1 for some t ∈ G.

Can take t = s−1: sU1 ∈ p ⇒ (sU1)−1 = U1s−1 ∈ p−1 U1s−1 = s−1sU1s−1 = s−1U′

1

dq(U1) = (dp−1 ◦ dp)(U1) = dp−1(U′

s−1) = Us−1t−1 = U1

dq(Uh) = dq(hU1) = hdq(U1) = hU1 = Uh So dq is identity on A.

Newelski Topological dynamics of stable groups

Conclusion

Hence qn ∗ qn = qn and qn is the generic type of U1. So U1 = q.

Newelski Topological dynamics of stable groups

Conclusion

Hence qn ∗ qn = qn and qn is the generic type of U1. So U1 = q.

Newelski Topological dynamics of stable groups