Forcing and liberable groups Filippo Calderoni Master of Pure and Applied Logic Universitat de Barcelona calderonifilippo@gmail.com www.calderonifilippo.com 16th April 2014 Filippo Calderoni (UB) XXV incontro AILA 1/17
Outline Almost Freeness Filtrations and Γ-invariant Liberating groups Filippo Calderoni (UB) XXV incontro AILA 2/17
Some History “The modern era of set-theoretic methods in algebra can be said to have begun on July 11, 1973 when Saharon Shelah borrowed L´ aszl´ o Fuchs’ Infinite Abelian Groups from the Hebrew University Library.” (P. Eklof, A. Mekler) 1 In 1974, could prove that the Whitehead’s Problem is independent from ZFC. 1 P. Eklof, A. Mekler. Almost Free Modules . North Holland, 2002. Filippo Calderoni (UB) XXV incontro AILA 3/17
Some History “The modern era of set-theoretic methods in algebra can be said to have begun on July 11, 1973 when Saharon Shelah borrowed L´ aszl´ o Fuchs’ Infinite Abelian Groups from the Hebrew University Library.” (P. Eklof, A. Mekler) 1 In 1974, could prove that the Whitehead’s Problem is independent from ZFC. 1 P. Eklof, A. Mekler. Almost Free Modules . North Holland, 2002. Filippo Calderoni (UB) XXV incontro AILA 3/17
Almost Free Groups Definition An (abelian) group G is free if and only if it has a basis. Ex. � α<κ Z Definition A group G is torsion-free if and only if every finitely generated subgroup of G is free. Ex. ( Q , +) Definition (Fuchs 1958) A group G is ℵ 1 -free if and only if every countably generated subgroup of G is free. Filippo Calderoni (UB) XXV incontro AILA 4/17
Almost Free Groups Definition An (abelian) group G is free if and only if it has a basis. Ex. � α<κ Z Definition A group G is torsion-free if and only if every finitely generated subgroup of G is free. Ex. ( Q , +) Definition (Fuchs 1958) A group G is ℵ 1 -free if and only if every countably generated subgroup of G is free. Filippo Calderoni (UB) XXV incontro AILA 4/17
Almost Free Groups Definition An (abelian) group G is free if and only if it has a basis. Ex. � α<κ Z Definition A group G is torsion-free if and only if every finitely generated subgroup of G is free. Ex. ( Q , +) Definition (Fuchs 1958) A group G is ℵ 1 -free if and only if every countably generated subgroup of G is free. Filippo Calderoni (UB) XXV incontro AILA 4/17
Examples and basic facts Since every subgroup of a free subgroup is free, it is clear that G free ⇒ G ℵ 1 -free . However, G free �⇐ G ℵ 1 -free Baer-Specker Group The direct product of ω copies of Z , Z ω := � Z i <ω is ℵ 1 -free (Specker 1950) but it is not free (Baer 1937). Filippo Calderoni (UB) XXV incontro AILA 5/17
Examples and basic facts Since every subgroup of a free subgroup is free, it is clear that G free ⇒ G ℵ 1 -free . However, G free �⇐ G ℵ 1 -free Baer-Specker Group The direct product of ω copies of Z , Z ω := � Z i <ω is ℵ 1 -free (Specker 1950) but it is not free (Baer 1937). Filippo Calderoni (UB) XXV incontro AILA 5/17
We shall focus on almost free groups of cardinality ℵ 1 . Question Can we force any almost free group of cardinality ℵ 1 to be free? This is not possible in general. In fact, ZFC � “ Z ω is not free” . True Question When is it possible to force an almost free group of cardinality ℵ 1 to be free? Filippo Calderoni (UB) XXV incontro AILA 6/17
We shall focus on almost free groups of cardinality ℵ 1 . Question Can we force any almost free group of cardinality ℵ 1 to be free? This is not possible in general. In fact, ZFC � “ Z ω is not free” . True Question When is it possible to force an almost free group of cardinality ℵ 1 to be free? Filippo Calderoni (UB) XXV incontro AILA 6/17
We shall focus on almost free groups of cardinality ℵ 1 . Question Can we force any almost free group of cardinality ℵ 1 to be free? This is not possible in general. In fact, ZFC � “ Z ω is not free” . True Question When is it possible to force an almost free group of cardinality ℵ 1 to be free? Filippo Calderoni (UB) XXV incontro AILA 6/17
We shall focus on almost free groups of cardinality ℵ 1 . Question Can we force any almost free group of cardinality ℵ 1 to be free? This is not possible in general. In fact, ZFC � “ Z ω is not free” . True Question When is it possible to force an almost free group of cardinality ℵ 1 to be free? Filippo Calderoni (UB) XXV incontro AILA 6/17
Filtrations Definition Given a group G of cardinality ℵ 1 , a filtration of G is a sequence { G α : α ∈ ℵ 1 } of subgroups of G whose union is G and such that for all α, β < κ : G α is a countable subgroup of G ; if α < β , then G α ⊆ G β ; if γ is a limit ordinal, then G γ = � α<γ G α . Theorem A group G of cardinality ℵ 1 is ℵ 1 -free if and only if it has a filtration { G α : α < ℵ 1 } consisting of free groups. Filippo Calderoni (UB) XXV incontro AILA 7/17
Filtrations Definition Given a group G of cardinality ℵ 1 , a filtration of G is a sequence { G α : α ∈ ℵ 1 } of subgroups of G whose union is G and such that for all α, β < κ : G α is a countable subgroup of G ; if α < β , then G α ⊆ G β ; if γ is a limit ordinal, then G γ = � α<γ G α . Theorem A group G of cardinality ℵ 1 is ℵ 1 -free if and only if it has a filtration { G α : α < ℵ 1 } consisting of free groups. Filippo Calderoni (UB) XXV incontro AILA 7/17
Γ-invariant Theorem (Eklof 1977) Let G be an ℵ 1 -free group of cardinality ℵ 1 . Then, G is free if and only if E := { α < ℵ 1 : G / G α is not ℵ 1 -free } is not stationary, for some filtration { G α : α < ℵ 1 } . This criterion does NOT depend on the filtration! In fact, any two filtrations agree on a club. Definition [ E ] in P ( ℵ 1 ) / club is the so called Γ -invariant of G . Filippo Calderoni (UB) XXV incontro AILA 8/17
Γ-invariant Theorem (Eklof 1977) Let G be an ℵ 1 -free group of cardinality ℵ 1 . Then, G is free if and only if E := { α < ℵ 1 : G / G α is not ℵ 1 -free } is not stationary, for some filtration { G α : α < ℵ 1 } . This criterion does NOT depend on the filtration! In fact, any two filtrations agree on a club. Definition [ E ] in P ( ℵ 1 ) / club is the so called Γ -invariant of G . Filippo Calderoni (UB) XXV incontro AILA 8/17
Γ-invariant Theorem (Eklof 1977) Let G be an ℵ 1 -free group of cardinality ℵ 1 . Then, G is free if and only if E := { α < ℵ 1 : G / G α is not ℵ 1 -free } is not stationary, for some filtration { G α : α < ℵ 1 } . This criterion does NOT depend on the filtration! In fact, any two filtrations agree on a club. Definition [ E ] in P ( ℵ 1 ) / club is the so called Γ -invariant of G . Filippo Calderoni (UB) XXV incontro AILA 8/17
Proof. ⇒ ) Let { x α : α < ℵ 1 } be a basis for G . Define the filtration { G α : α < ℵ 1 } such that G α = � x ξ : ξ < α � , then it turns out that E = ∅ . ⇐ ) Let { G α : α < ℵ 1 } be any filtration of G and assume that E is not stationary. That is, there exists a club C ⊆ ω 1 such that G / G α is ℵ 1 -free, for each α ∈ C . Moreover, { G α : α ∈ C } is a filtration of G . Claim : given any function f : ℵ 1 → C enumerating C , G f ( α +1) / G f ( α ) is free, for every α < ℵ 1 . We conclude that G is free because � G = G f ( α +1) / G f ( α ) . α ∈ℵ 1 Filippo Calderoni (UB) XXV incontro AILA 9/17
Proof. ⇒ ) Let { x α : α < ℵ 1 } be a basis for G . Define the filtration { G α : α < ℵ 1 } such that G α = � x ξ : ξ < α � , then it turns out that E = ∅ . ⇐ ) Let { G α : α < ℵ 1 } be any filtration of G and assume that E is not stationary. That is, there exists a club C ⊆ ω 1 such that G / G α is ℵ 1 -free, for each α ∈ C . Moreover, { G α : α ∈ C } is a filtration of G . Claim : given any function f : ℵ 1 → C enumerating C , G f ( α +1) / G f ( α ) is free, for every α < ℵ 1 . We conclude that G is free because � G = G f ( α +1) / G f ( α ) . α ∈ℵ 1 Filippo Calderoni (UB) XXV incontro AILA 9/17
Proof. ⇒ ) Let { x α : α < ℵ 1 } be a basis for G . Define the filtration { G α : α < ℵ 1 } such that G α = � x ξ : ξ < α � , then it turns out that E = ∅ . ⇐ ) Let { G α : α < ℵ 1 } be any filtration of G and assume that E is not stationary. That is, there exists a club C ⊆ ω 1 such that G / G α is ℵ 1 -free, for each α ∈ C . Moreover, { G α : α ∈ C } is a filtration of G . Claim : given any function f : ℵ 1 → C enumerating C , G f ( α +1) / G f ( α ) is free, for every α < ℵ 1 . We conclude that G is free because � G = G f ( α +1) / G f ( α ) . α ∈ℵ 1 Filippo Calderoni (UB) XXV incontro AILA 9/17
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