The relation of embeddability between uncountable torsion-free abelian groups Filippo Calderoni University of Turin March 14, 2017 1/22
Borel reducibility In the framework of (classical) Borel reducibility, relations are defined over Polish or standard Borel spaces. 2/22
Borel reducibility In the framework of (classical) Borel reducibility, relations are defined over Polish or standard Borel spaces. Definition Let P and Q be quasi-orders X and Y , respectively. We say that P Borel reduces to Q (or P ≤ B Q ) if and only if there is a Borel f : X → Y function such that x 1 P x 2 ⇔ f ( x 1 ) Q f ( x 2 ) . 2/22
Borel reducibility In the framework of (classical) Borel reducibility, relations are defined over Polish or standard Borel spaces. Definition Let P and Q be quasi-orders X and Y , respectively. We say that P Borel reduces to Q (or P ≤ B Q ) if and only if there is a Borel f : X → Y function such that x 1 P x 2 ⇔ f ( x 1 ) Q f ( x 2 ) . If E , F are equivalence relation and E ≤ B F we say roughly that E is not more complicated than F . 2/22
Analytic equivalence relations E Σ 1 • 1 • E ℓ ∞ • E G ∞ • E 1 • E S ∞ • id 2 <ω 1 • E ctble Polish group actions • E ∞ classifiable by N -structures countable Borel • E 0 • id R Σ 1 1 equivalence relation 3/22
Analytic equivalence relations E Σ 1 • 1 E Σ 1 1 is defined as the • E ℓ ∞ ≤ B -maximum and called the complete Σ 1 1 equiv- • E G ∞ • E 1 alence relation . • E S ∞ • id 2 <ω 1 • E ctble • E ∞ • E 0 • id R Σ 1 1 equivalence relation 3/22
Complete Σ 1 1 quasi-orders Definition Q is a complete Σ 1 1 quasi-order if and only if Q is Σ 1 1 and P ≤ B Q , for every Σ 1 1 quasi-order P . 4/22
Complete Σ 1 1 quasi-orders Definition Q is a complete Σ 1 1 quasi-order if and only if Q is Σ 1 1 and P ≤ B Q , for every Σ 1 1 quasi-order P . 1 quasi-order, then E Q := Q ∩ Q − 1 is a If Q is a complete Σ 1 complete Σ 1 1 equivalence relation. 4/22
Complete Σ 1 1 quasi-orders Definition Q is a complete Σ 1 1 quasi-order if and only if Q is Σ 1 1 and P ≤ B Q , for every Σ 1 1 quasi-order P . 1 quasi-order, then E Q := Q ∩ Q − 1 is a If Q is a complete Σ 1 complete Σ 1 1 equivalence relation. Theorem (Louveau-Rosendal 2005) The relation of embeddability between countable graphs ⊑ GRAPHS is a complete Σ 1 1 quasi-order. Thus the bi-embeddability ≡ GRAPHS is a complete Σ 1 1 equivalence relation. 4/22
Embeddability between coutable structures Definition Denote by X GRAPHS the space of countable (undirected) graphs with domain N . It is a closed subset of 2 N 2 , thus a Polish space. 5/22
Embeddability between coutable structures Definition Denote by X GRAPHS the space of countable (undirected) graphs with domain N . It is a closed subset of 2 N 2 , thus a Polish space. def ∃ h : N 1 − 1 T ⊑ GRAPHS V ⇐ ⇒ − − → N h is an isomorphism from T to V ↾ Im ( h ). 5/22
Embeddability between coutable structures Definition Denote by X GRAPHS the space of countable (undirected) graphs with domain N . It is a closed subset of 2 N 2 , thus a Polish space. def ∃ h : N 1 − 1 T ⊑ GRAPHS V ⇐ ⇒ − − → N h is an isomorphism from T to V ↾ Im ( h ). ⊑ GRAPHS is a Σ 1 1 quasi-order. we can consider the relation of embeddability on any space of countable structures (graphs, groups, etc...); 5/22
Embeddability of countable groups Theorem (Williams 2014) The embeddability between countable groups ⊑ GROUPS is a complete Σ 1 1 quasi-order. Williams defines a reduction from ⊑ GRAPHS to ⊑ GROUPS . 6/22
Embeddability of countable groups Theorem (Williams 2014) The embeddability between countable groups ⊑ GROUPS is a complete Σ 1 1 quasi-order. Williams defines a reduction from ⊑ GRAPHS to ⊑ GROUPS . The groups Williams builds to reduce ⊑ GRAPHS to ⊑ GROUPS are nonabelian. 6/22
Embeddability on abelian groups Question How about the embeddability between countable abelian groups? 7/22
Embeddability on abelian groups Question How about the embeddability between countable abelian groups? And the more challenging... Question How about the embeddability between countable torsion-free abelian groups? Is it a complete Σ 1 1 quasi-order? 7/22
What I’m not talking about today A few weeks ago... Theorem (C.-Thomas) The relation of embeddability ⊑ TFA between countable torsion-free abelian groups is a complete Σ 1 1 quasi-order. An old problem Is the isomorphism between countable torsion-free abelian groups S ∞ -complete (Borel complete)? 8/22
The “generalized” result A few more weeks ago... Theorem (C.) If κ is uncountable such that κ <κ = κ , then the relation ⊑ κ TFA of embeddability between TFA groups of size κ is a complete Σ 1 1 quasi-order. 9/22
The “generalized” result A few more weeks ago... Theorem (C.) If κ is uncountable such that κ <κ = κ , then the relation ⊑ κ TFA of embeddability between TFA groups of size κ is a complete Σ 1 1 quasi-order. Generalized Descriptive Set Theory is not the mere generalization of Classical Descriptive Set Theory. 9/22
The “generalized” result A few more weeks ago... Theorem (C.) If κ is uncountable such that κ <κ = κ , then the relation ⊑ κ TFA of embeddability between TFA groups of size κ is a complete Σ 1 1 quasi-order. Generalized Descriptive Set Theory is not the mere generalization of Classical Descriptive Set Theory. The proof relies on the existence of an almost-full embedding G : Graphs → Ab (Prze´ zdziecki 2014). 9/22
Generalied Descriptive Set Theory Let κ be an uncountable cardinal. Definition The generalized Baire space on κ is κ κ := { x | x : κ → κ } endowed with the ( bounded ) topology , i.e. the one generated by the sets of the form N s := { x ∈ κ κ | x ⊇ s } , where s ∈ <κ κ . 10/22
Generalied Descriptive Set Theory Let κ be an uncountable cardinal. Definition The generalized Baire space on κ is κ κ := { x | x : κ → κ } endowed with the ( bounded ) topology , i.e. the one generated by the sets of the form N s := { x ∈ κ κ | x ⊇ s } , where s ∈ <κ κ . The generalized Cantor space is the closed subspace κ 2 of κ κ consisting of the binary sequences. 10/22
Generalied Descriptive Set Theory Let κ be an uncountable cardinal. Definition The generalized Baire space on κ is κ κ := { x | x : κ → κ } endowed with the ( bounded ) topology , i.e. the one generated by the sets of the form N s := { x ∈ κ κ | x ⊇ s } , where s ∈ <κ κ . The generalized Cantor space is the closed subspace κ 2 of κ κ consisting of the binary sequences. They are not metrizable (unless cof ( κ ) = ω ), and their density character is κ <κ and 2 <κ . 10/22
The most common approach We assume κ <κ = κ κ κ is a κ -space (i.e., it has a base of size κ ). κ + - Borel sets are the closure under complements and ≤ κ -unions of the basic open sets. They form a proper subset of P ( κ κ ) that can be stratified in a hierarchy with exactly κ + -many levels (the Σ 0 α ’s and Π 0 α ’s). A κ -space is standard Borel if it is κ + -Borel isomorphic to a κ + -Borel subset of κ κ . Let X be standard Borel. Then a set A ⊆ X is ( κ -) analytic or Σ 1 1 if it is a projection of a closed subset of κ κ × X . 11/22
Generalized Borel reducibility We denote by X κ GRAPHS , X κ TFA the standard Borel κ -spaces of the respective structures over κ . TFA the Σ 1 We denote by ⊑ κ GRAPHS , ⊑ κ 1 quasi-orders of embeddability on the respective spaces. 12/22
Generalized Borel reducibility We denote by X κ GRAPHS , X κ TFA the standard Borel κ -spaces of the respective structures over κ . TFA the Σ 1 We denote by ⊑ κ GRAPHS , ⊑ κ 1 quasi-orders of embeddability on the respective spaces. Definition Let X and Y be standard Borel κ -space, and P , Q be binary relations over X and Y , respectively. We say that P Borel reduces to Q (or P ≤ B Q ) if and only if there is a κ + -Borel f : X → Y such that x 1 P x 2 ⇔ f ( x 1 ) Q f ( x 2 ) . The definition of Σ 1 1 -completeness is translated verbatim. 12/22
Generalized Louveau-Rosendal Theorem Theorem (Motto Ros 2013) GRAPHS is a complete Σ 1 If κ is weakly compact, then ⊑ κ 1 quasi-order. 13/22
Generalized Louveau-Rosendal Theorem Theorem (Motto Ros 2013) GRAPHS is a complete Σ 1 If κ is weakly compact, then ⊑ κ 1 quasi-order. Theorem (Mildenberger-Motto Ros) If κ is uncountable such that κ <κ = κ , then ⊑ κ GRAPHS is a complete Σ 1 1 quasi-order. 13/22
Theorem (C.) If κ is uncountable such that κ <κ = κ , then the relation ⊑ κ TFA of embeddability between TFA groups of size κ is a complete Σ 1 1 quasi-order. Proof (sketch) It suffices to show that ⊑ κ GRAPHS ≤ B ⊑ κ TFA . Let Γ be a skeleton of the category Graph ctble Suppose that every graph in Γ has some subset of N as vertex set. Let W κ be a universal graph of size κ . I.e. every graph of size κ embeds into it. If T is a graph, then let [ T ] <ω 1 be the poset of countable induced subgraphs of T ordered by inclusion. 14/22
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