Definably quotients of locally definable groups Y. Peterzil (joint work with P . Eleftheriou) Department of Mathematics University of Haifa Oleron 2011 Y. Peterzil (University of Haifa) Definable quotients 1
Setting Let M = M 1 ⊔ M 2 be an arbitrary structure, a disjoint union of two sorts (no maps between M 1 and M 2 ). Problem What are the definable (interpretable) groups in M ? (say, in terms of M 1 and in M 2 ) Y. Peterzil (University of Haifa) Definable quotients 2
Setting Let M = M 1 ⊔ M 2 be an arbitrary structure, a disjoint union of two sorts (no maps between M 1 and M 2 ). Problem What are the definable (interpretable) groups in M ? (say, in terms of M 1 and in M 2 ) Y. Peterzil (University of Haifa) Definable quotients 2
Setting Let M = M 1 ⊔ M 2 be an arbitrary structure, a disjoint union of two sorts (no maps between M 1 and M 2 ). Problem What are the definable (interpretable) groups in M ? (say, in terms of M 1 and in M 2 ) Y. Peterzil (University of Haifa) Definable quotients 2
Answer 1 G = H 1 × H 2 , where H i is definable in M i , i = 1 , 2. Answer 2 G = ( H 1 × H 2 ) / F , where F is a finite subgroup. Answer 3 H 1 H 2 0 ✲ ✲ G ✲ ✲ 1 A central extension G of a definable group H 2 in M 2 by a definable group H 1 in M 1 (via, say a finite co-cycle σ : H 2 × H 2 → H 1 ). Y. Peterzil (University of Haifa) Definable quotients 3
Answer 1 G = H 1 × H 2 , where H i is definable in M i , i = 1 , 2. Answer 2 G = ( H 1 × H 2 ) / F , where F is a finite subgroup. Answer 3 H 1 H 2 0 ✲ ✲ G ✲ ✲ 1 A central extension G of a definable group H 2 in M 2 by a definable group H 1 in M 1 (via, say a finite co-cycle σ : H 2 × H 2 → H 1 ). Y. Peterzil (University of Haifa) Definable quotients 3
Answer 1 G = H 1 × H 2 , where H i is definable in M i , i = 1 , 2. Answer 2 G = ( H 1 × H 2 ) / F , where F is a finite subgroup. Answer 3 H 1 H 2 0 ✲ ✲ G ✲ ✲ 1 A central extension G of a definable group H 2 in M 2 by a definable group H 1 in M 1 (via, say a finite co-cycle σ : H 2 × H 2 → H 1 ). Y. Peterzil (University of Haifa) Definable quotients 3
One more answer G = ( H 1 × H 2 ) / Γ , where H i is a locally definable group in M i and Γ an infinite small, non-definable, subgroup. But G is definable in M 1 ⊔ M 2 ! Definition A locally definable group �G , ·� (in an ω -saturated structure) is a countable directed union of definable sets G = � n X n ⊆ M k , such that (i) for every m , n , the restriction of multiplication to X m × X n is definable (and (ii) for every m , n there exists ℓ with X m · X n ⊆ X ℓ , X − 1 ⊆ X ℓ ). n Example G definable group, e ∈ X ⊆ G a definable set, and G = � X � the subgroup of G generated by X . Y. Peterzil (University of Haifa) Definable quotients 4
One more answer G = ( H 1 × H 2 ) / Γ , where H i is a locally definable group in M i and Γ an infinite small, non-definable, subgroup. But G is definable in M 1 ⊔ M 2 ! Definition A locally definable group �G , ·� (in an ω -saturated structure) is a countable directed union of definable sets G = � n X n ⊆ M k , such that (i) for every m , n , the restriction of multiplication to X m × X n is definable (and (ii) for every m , n there exists ℓ with X m · X n ⊆ X ℓ , X − 1 ⊆ X ℓ ). n Example G definable group, e ∈ X ⊆ G a definable set, and G = � X � the subgroup of G generated by X . Y. Peterzil (University of Haifa) Definable quotients 4
One more answer G = ( H 1 × H 2 ) / Γ , where H i is a locally definable group in M i and Γ an infinite small, non-definable, subgroup. But G is definable in M 1 ⊔ M 2 ! Definition A locally definable group �G , ·� (in an ω -saturated structure) is a countable directed union of definable sets G = � n X n ⊆ M k , such that (i) for every m , n , the restriction of multiplication to X m × X n is definable (and (ii) for every m , n there exists ℓ with X m · X n ⊆ X ℓ , X − 1 ⊆ X ℓ ). n Example G definable group, e ∈ X ⊆ G a definable set, and G = � X � the subgroup of G generated by X . Y. Peterzil (University of Haifa) Definable quotients 4
One more answer G = ( H 1 × H 2 ) / Γ , where H i is a locally definable group in M i and Γ an infinite small, non-definable, subgroup. But G is definable in M 1 ⊔ M 2 ! Definition A locally definable group �G , ·� (in an ω -saturated structure) is a countable directed union of definable sets G = � n X n ⊆ M k , such that (i) for every m , n , the restriction of multiplication to X m × X n is definable (and (ii) for every m , n there exists ℓ with X m · X n ⊆ X ℓ , X − 1 ⊆ X ℓ ). n Example G definable group, e ∈ X ⊆ G a definable set, and G = � X � the subgroup of G generated by X . Y. Peterzil (University of Haifa) Definable quotients 4
Definable quotients Definition For Γ ⊆ G , we say that G / Γ is definable (interpretable) if there exists a definable (interpretable) group G and a locally definable surjective homomorphism φ : G → G . Example � R , <, + , a � a large ordered, divisible, abelian group. Take G = � n ( − na , na ) , a locally definable subgroup. Γ = Z a ⊆ G . Then G / Γ is definable: There is a locally definable surjection φ : G → � [ − a , a ] , + mod a � . Y. Peterzil (University of Haifa) Definable quotients 5
Definable quotients Definition For Γ ⊆ G , we say that G / Γ is definable (interpretable) if there exists a definable (interpretable) group G and a locally definable surjective homomorphism φ : G → G . Example � R , <, + , a � a large ordered, divisible, abelian group. Take G = � n ( − na , na ) , a locally definable subgroup. Γ = Z a ⊆ G . Then G / Γ is definable: There is a locally definable surjection φ : G → � [ − a , a ] , + mod a � . Y. Peterzil (University of Haifa) Definable quotients 5
M is an arbitrary κ -saturated structure. Fact For G a locally definable group, and Γ � G a small normal subgroup. The group G / Γ is definable (interpretable) in M iff there exists a definable X ⊆ G such that (i) G = Γ · X (ii) X ∩ Γ is finite. (The group Γ is “a lattice in G ” and the set X is a “fundamental set” ) . Proof IF : We assume G = Γ X . The set XX − 1 definable ⇒ XX − 1 ⊆ FX (for finite F ⊆ Γ ) ⇒ XX − 1 ∩ Γ ⊆ FX ∩ Γ = F ( X ∩ Γ) is finite. ⇒ ‘ x 1 Γ = x 2 Γ ′ is definable for x 1 , x 2 ∈ X . Similarly, the relation x 1 x 2 Γ = x 3 Γ is definable for x 1 , x 2 , x 3 ∈ X . ⇒ can define a group on X / Γ ( ∼ = G / Γ ). Y. Peterzil (University of Haifa) Definable quotients 6
M is an arbitrary κ -saturated structure. Fact For G a locally definable group, and Γ � G a small normal subgroup. The group G / Γ is definable (interpretable) in M iff there exists a definable X ⊆ G such that (i) G = Γ · X (ii) X ∩ Γ is finite. (The group Γ is “a lattice in G ” and the set X is a “fundamental set” ) . Proof IF : We assume G = Γ X . The set XX − 1 definable ⇒ XX − 1 ⊆ FX (for finite F ⊆ Γ ) ⇒ XX − 1 ∩ Γ ⊆ FX ∩ Γ = F ( X ∩ Γ) is finite. ⇒ ‘ x 1 Γ = x 2 Γ ′ is definable for x 1 , x 2 ∈ X . Similarly, the relation x 1 x 2 Γ = x 3 Γ is definable for x 1 , x 2 , x 3 ∈ X . ⇒ can define a group on X / Γ ( ∼ = G / Γ ). Y. Peterzil (University of Haifa) Definable quotients 6
M is an arbitrary κ -saturated structure. Fact For G a locally definable group, and Γ � G a small normal subgroup. The group G / Γ is definable (interpretable) in M iff there exists a definable X ⊆ G such that (i) G = Γ · X (ii) X ∩ Γ is finite. (The group Γ is “a lattice in G ” and the set X is a “fundamental set” ) . Proof IF : We assume G = Γ X . The set XX − 1 definable ⇒ XX − 1 ⊆ FX (for finite F ⊆ Γ ) ⇒ XX − 1 ∩ Γ ⊆ FX ∩ Γ = F ( X ∩ Γ) is finite. ⇒ ‘ x 1 Γ = x 2 Γ ′ is definable for x 1 , x 2 ∈ X . Similarly, the relation x 1 x 2 Γ = x 3 Γ is definable for x 1 , x 2 , x 3 ∈ X . ⇒ can define a group on X / Γ ( ∼ = G / Γ ). Y. Peterzil (University of Haifa) Definable quotients 6
M is an arbitrary κ -saturated structure. Fact For G a locally definable group, and Γ � G a small normal subgroup. The group G / Γ is definable (interpretable) in M iff there exists a definable X ⊆ G such that (i) G = Γ · X (ii) X ∩ Γ is finite. (The group Γ is “a lattice in G ” and the set X is a “fundamental set” ) . Proof IF : We assume G = Γ X . The set XX − 1 definable ⇒ XX − 1 ⊆ FX (for finite F ⊆ Γ ) ⇒ XX − 1 ∩ Γ ⊆ FX ∩ Γ = F ( X ∩ Γ) is finite. ⇒ ‘ x 1 Γ = x 2 Γ ′ is definable for x 1 , x 2 ∈ X . Similarly, the relation x 1 x 2 Γ = x 3 Γ is definable for x 1 , x 2 , x 3 ∈ X . ⇒ can define a group on X / Γ ( ∼ = G / Γ ). Y. Peterzil (University of Haifa) Definable quotients 6
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