Homogeneity and algebraic closure in free...
Transcript of Homogeneity and algebraic closure in free...
![Page 1: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/1.jpg)
Homogeneity and algebraic closure in free groups
Abderezak Ould Houcine
Universite de Mons, Universite Lyon 1
Equations and First-order properties in Groups
Montreal, 15 october 2010
![Page 2: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/2.jpg)
Contents
1 Homogeneity & prime modelsDefinitionsExistential homogeneity & prime modelsHomogeneity
2 Algebraic & definable closureDefinitionsConstructibility over the algebraic closureA counterexample
![Page 3: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/3.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Contents
1 Homogeneity & prime modelsDefinitionsExistential homogeneity & prime modelsHomogeneity
2 Algebraic & definable closureDefinitionsConstructibility over the algebraic closureA counterexample
![Page 4: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/4.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Definitions
Homogeneity & existential homogeneity
![Page 5: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/5.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Definitions
Homogeneity & existential homogeneity
Let M be a countable model.
![Page 6: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/6.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Definitions
Homogeneity & existential homogeneity
Let M be a countable model.
Let a ∈ Mn.
![Page 7: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/7.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Definitions
Homogeneity & existential homogeneity
Let M be a countable model.
Let a ∈ Mn. The type of a is defined by
![Page 8: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/8.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Definitions
Homogeneity & existential homogeneity
Let M be a countable model.
Let a ∈ Mn. The type of a is defined by
tpM(a) = {ψ(x)|M |= ψ(a)}.
![Page 9: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/9.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Definitions
Homogeneity & existential homogeneity
Let M be a countable model.
Let a ∈ Mn. The type of a is defined by
tpM(a) = {ψ(x)|M |= ψ(a)}.
M is homogeneous
![Page 10: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/10.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Definitions
Homogeneity & existential homogeneity
Let M be a countable model.
Let a ∈ Mn. The type of a is defined by
tpM(a) = {ψ(x)|M |= ψ(a)}.
M is homogeneous ⇔ for any n ≥ 1, for any a, b ∈ Mn,
![Page 11: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/11.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Definitions
Homogeneity & existential homogeneity
Let M be a countable model.
Let a ∈ Mn. The type of a is defined by
tpM(a) = {ψ(x)|M |= ψ(a)}.
M is homogeneous ⇔ for any n ≥ 1, for any a, b ∈ Mn,tpM(a) = tpM(b) =⇒ ∃f ∈ Aut(M) s.t. f (a) = f (b).
![Page 12: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/12.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Definitions
Homogeneity & existential homogeneity
Let M be a countable model.
Let a ∈ Mn. The type of a is defined by
tpM(a) = {ψ(x)|M |= ψ(a)}.
M is homogeneous ⇔ for any n ≥ 1, for any a, b ∈ Mn,tpM(a) = tpM(b) =⇒ ∃f ∈ Aut(M) s.t. f (a) = f (b).
M is ∃-homogeneous
![Page 13: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/13.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Definitions
Homogeneity & existential homogeneity
Let M be a countable model.
Let a ∈ Mn. The type of a is defined by
tpM(a) = {ψ(x)|M |= ψ(a)}.
M is homogeneous ⇔ for any n ≥ 1, for any a, b ∈ Mn,tpM(a) = tpM(b) =⇒ ∃f ∈ Aut(M) s.t. f (a) = f (b).
M is ∃-homogeneous ⇔ for any n ≥ 1, for any a, b ∈ Mn,
![Page 14: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/14.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Definitions
Homogeneity & existential homogeneity
Let M be a countable model.
Let a ∈ Mn. The type of a is defined by
tpM(a) = {ψ(x)|M |= ψ(a)}.
M is homogeneous ⇔ for any n ≥ 1, for any a, b ∈ Mn,tpM(a) = tpM(b) =⇒ ∃f ∈ Aut(M) s.t. f (a) = f (b).
M is ∃-homogeneous ⇔ for any n ≥ 1, for any a, b ∈ Mn,tpM
∃ (a) = tpM∃ (b) =⇒ ∃f ∈ Aut(M) s.t. f (a) = f (b).
![Page 15: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/15.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Definitions
Homogeneity & existential homogeneity
Let M be a countable model.
Let a ∈ Mn. The type of a is defined by
tpM(a) = {ψ(x)|M |= ψ(a)}.
M is homogeneous ⇔ for any n ≥ 1, for any a, b ∈ Mn,tpM(a) = tpM(b) =⇒ ∃f ∈ Aut(M) s.t. f (a) = f (b).
M is ∃-homogeneous ⇔ for any n ≥ 1, for any a, b ∈ Mn,tpM
∃ (a) = tpM∃ (b) =⇒ ∃f ∈ Aut(M) s.t. f (a) = f (b).
Remark.
![Page 16: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/16.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Definitions
Homogeneity & existential homogeneity
Let M be a countable model.
Let a ∈ Mn. The type of a is defined by
tpM(a) = {ψ(x)|M |= ψ(a)}.
M is homogeneous ⇔ for any n ≥ 1, for any a, b ∈ Mn,tpM(a) = tpM(b) =⇒ ∃f ∈ Aut(M) s.t. f (a) = f (b).
M is ∃-homogeneous ⇔ for any n ≥ 1, for any a, b ∈ Mn,tpM
∃ (a) = tpM∃ (b) =⇒ ∃f ∈ Aut(M) s.t. f (a) = f (b).
Remark. ∃-homogeneity =⇒ homogeneity.
![Page 17: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/17.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Definitions
Prime Models
![Page 18: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/18.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Definitions
Prime Models
M is called prime
![Page 19: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/19.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Definitions
Prime Models
M is called prime if M is elementary embeddable in everymodel of Th(M).
![Page 20: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/20.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
The free group of rank 2
![Page 21: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/21.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
The free group of rank 2
Theorem 1 (A. Nies, 2003)
![Page 22: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/22.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
The free group of rank 2
Theorem 1 (A. Nies, 2003)
A free group F2 of rank 2 is ∃-homogeneous and not prime.
![Page 23: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/23.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
The free group of rank 2
Theorem 1 (A. Nies, 2003)
A free group F2 of rank 2 is ∃-homogeneous and not prime. Inparticular Th(F2) has no prime model.
![Page 24: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/24.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
The free group of rank 2
Theorem 1 (A. Nies, 2003)
A free group F2 of rank 2 is ∃-homogeneous and not prime. Inparticular Th(F2) has no prime model.
The proof uses the following strong property of the free group F2
with basis {a, b}:
![Page 25: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/25.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
The free group of rank 2
Theorem 1 (A. Nies, 2003)
A free group F2 of rank 2 is ∃-homogeneous and not prime. Inparticular Th(F2) has no prime model.
The proof uses the following strong property of the free group F2
with basis {a, b}: there exists a quantifier-free formula ϕ(x , y),
![Page 26: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/26.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
The free group of rank 2
Theorem 1 (A. Nies, 2003)
A free group F2 of rank 2 is ∃-homogeneous and not prime. Inparticular Th(F2) has no prime model.
The proof uses the following strong property of the free group F2
with basis {a, b}: there exists a quantifier-free formula ϕ(x , y),such that for any endomorphism f of F2 if F2 |= ϕ(f (a), f (b)) thenf is an embedding.
![Page 27: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/27.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
The free group of rank 2
Theorem 1 (A. Nies, 2003)
A free group F2 of rank 2 is ∃-homogeneous and not prime. Inparticular Th(F2) has no prime model.
The proof uses the following strong property of the free group F2
with basis {a, b}: there exists a quantifier-free formula ϕ(x , y),such that for any endomorphism f of F2 if F2 |= ϕ(f (a), f (b)) thenf is an embedding.
Indeed, we can take ϕ(x , y) := [x , y ] 6= 1
![Page 28: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/28.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
![Page 29: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/29.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
Question:
![Page 30: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/30.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
Question: What can be said about the ∃-homogeneity and”primeness” of two-generated torsion-free hyperbolic groups?
![Page 31: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/31.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
Question: What can be said about the ∃-homogeneity and”primeness” of two-generated torsion-free hyperbolic groups?
Definition
A group G is said to be co-hopfian, if any injective endomorphismof G is an automorphism.
![Page 32: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/32.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
Question: What can be said about the ∃-homogeneity and”primeness” of two-generated torsion-free hyperbolic groups?
Definition
A group G is said to be co-hopfian, if any injective endomorphismof G is an automorphism.
That is a group is co-hopfian if it does not contain a subgroupisomorphic to itself.
![Page 33: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/33.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
![Page 34: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/34.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
Examples:
![Page 35: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/35.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
Examples:
Finite groups, the group of the rationals Q.
![Page 36: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/36.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
Examples:
Finite groups, the group of the rationals Q.
SLn(Z) with n ≥ 3 (G. Prasad, 1976).
![Page 37: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/37.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
Examples:
Finite groups, the group of the rationals Q.
SLn(Z) with n ≥ 3 (G. Prasad, 1976).
Nonabelian freely indecomposable trosion-free hyperbolicgroups (Z. Sela, 1997).
![Page 38: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/38.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
Examples:
Finite groups, the group of the rationals Q.
SLn(Z) with n ≥ 3 (G. Prasad, 1976).
Nonabelian freely indecomposable trosion-free hyperbolicgroups (Z. Sela, 1997).
Mapping class groups of closed surfaces (N.V. Ivanov, J.D.McCarthy, 1999).
![Page 39: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/39.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
Examples:
Finite groups, the group of the rationals Q.
SLn(Z) with n ≥ 3 (G. Prasad, 1976).
Nonabelian freely indecomposable trosion-free hyperbolicgroups (Z. Sela, 1997).
Mapping class groups of closed surfaces (N.V. Ivanov, J.D.McCarthy, 1999).
Bn/Z (Bn), where Bn is the Braid group on n ≥ 4 strands(R.W. Bell, D. Margalit, 2005).
![Page 40: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/40.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
Examples:
Finite groups, the group of the rationals Q.
SLn(Z) with n ≥ 3 (G. Prasad, 1976).
Nonabelian freely indecomposable trosion-free hyperbolicgroups (Z. Sela, 1997).
Mapping class groups of closed surfaces (N.V. Ivanov, J.D.McCarthy, 1999).
Bn/Z (Bn), where Bn is the Braid group on n ≥ 4 strands(R.W. Bell, D. Margalit, 2005).
Out(Fn), where Fn is a free group of rank n (B. Farb, M.Handel, 2007).
![Page 41: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/41.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
![Page 42: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/42.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
We introduce a strong form of the co-hopf property.
![Page 43: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/43.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
We introduce a strong form of the co-hopf property.
Definition
![Page 44: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/44.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
We introduce a strong form of the co-hopf property.
Definition
A group G is said to be strongly co-hopfian,
![Page 45: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/45.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
We introduce a strong form of the co-hopf property.
Definition
A group G is said to be strongly co-hopfian, if there exists afinite subset S ⊆ G \ {1}
![Page 46: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/46.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
We introduce a strong form of the co-hopf property.
Definition
A group G is said to be strongly co-hopfian, if there exists afinite subset S ⊆ G \ {1} such that for any endomorphism ϕ of G ,if 1 6∈ ϕ(S) then ϕ is an automorphism.
![Page 47: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/47.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
We introduce a strong form of the co-hopf property.
Definition
A group G is said to be strongly co-hopfian, if there exists afinite subset S ⊆ G \ {1} such that for any endomorphism ϕ of G ,if 1 6∈ ϕ(S) then ϕ is an automorphism.
Examples:
![Page 48: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/48.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
We introduce a strong form of the co-hopf property.
Definition
A group G is said to be strongly co-hopfian, if there exists afinite subset S ⊆ G \ {1} such that for any endomorphism ϕ of G ,if 1 6∈ ϕ(S) then ϕ is an automorphism.
Examples:
Finite groups, the group of the rationals Q.
![Page 49: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/49.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
We introduce a strong form of the co-hopf property.
Definition
A group G is said to be strongly co-hopfian, if there exists afinite subset S ⊆ G \ {1} such that for any endomorphism ϕ of G ,if 1 6∈ ϕ(S) then ϕ is an automorphism.
Examples:
Finite groups, the group of the rationals Q.
Tarski Monster groups.
![Page 50: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/50.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
![Page 51: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/51.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
The previous notion is very interesting from the view point ofmodel theory:
![Page 52: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/52.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
The previous notion is very interesting from the view point ofmodel theory:
Lemma
![Page 53: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/53.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
The previous notion is very interesting from the view point ofmodel theory:
Lemma
A finitely presented strongly co-hopfian group is ∃-homogeneousand prime.
![Page 54: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/54.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
The previous notion is very interesting from the view point ofmodel theory:
Lemma
A finitely presented strongly co-hopfian group is ∃-homogeneousand prime.
Let G = 〈a|r1(a), r2(a), . . . , rn(a)〉 be finitely presented andstrongly co-hopfian.
![Page 55: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/55.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
The previous notion is very interesting from the view point ofmodel theory:
Lemma
A finitely presented strongly co-hopfian group is ∃-homogeneousand prime.
Let G = 〈a|r1(a), r2(a), . . . , rn(a)〉 be finitely presented andstrongly co-hopfian. Then there exists a quantifier-free formulaϕ(x) such that for any endomorphism f of G , if G |= ϕ(f (a)) thenf is an automorphism.
![Page 56: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/56.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
The previous notion is very interesting from the view point ofmodel theory:
Lemma
A finitely presented strongly co-hopfian group is ∃-homogeneousand prime.
Let G = 〈a|r1(a), r2(a), . . . , rn(a)〉 be finitely presented andstrongly co-hopfian. Then there exists a quantifier-free formulaϕ(x) such that for any endomorphism f of G , if G |= ϕ(f (a)) thenf is an automorphism. A property analogous to that of the freegroup of rank 2.
![Page 57: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/57.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
![Page 58: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/58.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
Theorem 2
![Page 59: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/59.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
Theorem 2
A non-free two-generated torsion-free hyperbolic group is stronglyco-hopfian.
![Page 60: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/60.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
![Page 61: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/61.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
Since torsion-free hyperbolic groups are finitely presented, weconclude by the previous Lemma:
![Page 62: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/62.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
Since torsion-free hyperbolic groups are finitely presented, weconclude by the previous Lemma:
Theorem 3
![Page 63: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/63.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Two-generated torsion-free hyperbolic groups
Since torsion-free hyperbolic groups are finitely presented, weconclude by the previous Lemma:
Theorem 3
A non-free two-generated torsion-free hyperbolic group is∃-homogeneous and prime.
![Page 64: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/64.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 2
![Page 65: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/65.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 2
Let Γ be a non-free two-generated trosion-free hyperbolic group.
![Page 66: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/66.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 2
Let Γ be a non-free two-generated trosion-free hyperbolic group.
There exists a sequence of subroups Γ = Γ1 ≥ Γ2 ≥ · · · ≥ Γn
satisfying the following properties:
![Page 67: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/67.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 2
Let Γ be a non-free two-generated trosion-free hyperbolic group.
There exists a sequence of subroups Γ = Γ1 ≥ Γ2 ≥ · · · ≥ Γn
satisfying the following properties:
(i) Each Γi is two-generated, hyperbolic and quasiconvex;
![Page 68: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/68.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 2
Let Γ be a non-free two-generated trosion-free hyperbolic group.
There exists a sequence of subroups Γ = Γ1 ≥ Γ2 ≥ · · · ≥ Γn
satisfying the following properties:
(i) Each Γi is two-generated, hyperbolic and quasiconvex;(ii) Γi = 〈Γi+1, t|A
t = B〉, where A and B are a nontrivialmalnormal cyclic subgroups of Γi+1;
![Page 69: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/69.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 2
Let Γ be a non-free two-generated trosion-free hyperbolic group.
There exists a sequence of subroups Γ = Γ1 ≥ Γ2 ≥ · · · ≥ Γn
satisfying the following properties:
(i) Each Γi is two-generated, hyperbolic and quasiconvex;(ii) Γi = 〈Γi+1, t|A
t = B〉, where A and B are a nontrivialmalnormal cyclic subgroups of Γi+1;(iii) Γn is a rigid subgroup of Γ.
![Page 70: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/70.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 2
![Page 71: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/71.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 2
Γn is Γ-determined;
![Page 72: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/72.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 2
Γn is Γ-determined; that is, there exists a finite subsetS ⊆ G \ {1} such that for any homomorphism ϕ : Γn → Γ, if1 6∈ ϕ(S) then ϕ is an embedding.
![Page 73: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/73.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 2
Γn is Γ-determined; that is, there exists a finite subsetS ⊆ G \ {1} such that for any homomorphism ϕ : Γn → Γ, if1 6∈ ϕ(S) then ϕ is an embedding.
Let ϕ be an endomorphism of Γ such that 1 6∈ ϕ(S). Thenthe restriction of ϕ to every Γi is an automorphism of Γi .
![Page 74: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/74.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 2
Γn is Γ-determined; that is, there exists a finite subsetS ⊆ G \ {1} such that for any homomorphism ϕ : Γn → Γ, if1 6∈ ϕ(S) then ϕ is an embedding.
Let ϕ be an endomorphism of Γ such that 1 6∈ ϕ(S). Thenthe restriction of ϕ to every Γi is an automorphism of Γi . Inparticular ϕ is an automorphism of Γ.
![Page 75: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/75.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Quasi-axiomatizable groups
![Page 76: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/76.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Quasi-axiomatizable groups
Definition (A. Nies)
![Page 77: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/77.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Quasi-axiomatizable groups
Definition (A. Nies)
A finitely generated group G is said to be Quasi-Axiomatizable
![Page 78: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/78.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Quasi-axiomatizable groups
Definition (A. Nies)
A finitely generated group G is said to be Quasi-Axiomatizable ifany finitely generated group which is elementary equivalent to G isisomorphic to G .
![Page 79: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/79.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Quasi-axiomatizable groups
Definition (A. Nies)
A finitely generated group G is said to be Quasi-Axiomatizable ifany finitely generated group which is elementary equivalent to G isisomorphic to G .
Question (A. Nies):
![Page 80: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/80.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Quasi-axiomatizable groups
Definition (A. Nies)
A finitely generated group G is said to be Quasi-Axiomatizable ifany finitely generated group which is elementary equivalent to G isisomorphic to G .
Question (A. Nies): Is there a f.g. group which is prime but notQA?
![Page 81: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/81.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Quasi-axiomatizable groups
![Page 82: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/82.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Quasi-axiomatizable groups
Using Sela’s work on the elementary theory of torsion-freehyperbolic groups, we have
![Page 83: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/83.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Quasi-axiomatizable groups
Using Sela’s work on the elementary theory of torsion-freehyperbolic groups, we have
Theorem 4
![Page 84: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/84.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Quasi-axiomatizable groups
Using Sela’s work on the elementary theory of torsion-freehyperbolic groups, we have
Theorem 4
Let Γ be a two-generated trosion-free hyperbolic group.
![Page 85: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/85.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Quasi-axiomatizable groups
Using Sela’s work on the elementary theory of torsion-freehyperbolic groups, we have
Theorem 4
Let Γ be a two-generated trosion-free hyperbolic group. Then Γ iselementary equivalent to Γ ∗ Z.
![Page 86: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/86.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Quasi-axiomatizable groups
Using Sela’s work on the elementary theory of torsion-freehyperbolic groups, we have
Theorem 4
Let Γ be a two-generated trosion-free hyperbolic group. Then Γ iselementary equivalent to Γ ∗ Z.
Hence
![Page 87: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/87.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Quasi-axiomatizable groups
Using Sela’s work on the elementary theory of torsion-freehyperbolic groups, we have
Theorem 4
Let Γ be a two-generated trosion-free hyperbolic group. Then Γ iselementary equivalent to Γ ∗ Z.
Hence if Γ is non-free then it is prime but not QA.
![Page 88: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/88.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Existential homogeneity in free groups
![Page 89: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/89.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Existential homogeneity in free groups
Question : What can be said about existential homogeneity infree groups?
![Page 90: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/90.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Existential homogeneity in free groups
Question : What can be said about existential homogeneity infree groups?Recall that:
![Page 91: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/91.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Existential homogeneity in free groups
Question : What can be said about existential homogeneity infree groups?Recall that:
Definition
![Page 92: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/92.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Existential homogeneity in free groups
Question : What can be said about existential homogeneity infree groups?Recall that:
Definition
Let M be a model and N a submodel of M.
![Page 93: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/93.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Existential homogeneity in free groups
Question : What can be said about existential homogeneity infree groups?Recall that:
Definition
Let M be a model and N a submodel of M. The model N is saidto be existentially closed (abbreviated e.c.) in M,
![Page 94: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/94.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Existential homogeneity in free groups
Question : What can be said about existential homogeneity infree groups?Recall that:
Definition
Let M be a model and N a submodel of M. The model N is saidto be existentially closed (abbreviated e.c.) in M, if for anyexistential formula ϕ with parameters from N , if M |= ϕ, thenN |= ϕ.
![Page 95: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/95.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Existential homogeneity in free groups
Question : What can be said about existential homogeneity infree groups?Recall that:
Definition
Let M be a model and N a submodel of M. The model N is saidto be existentially closed (abbreviated e.c.) in M, if for anyexistential formula ϕ with parameters from N , if M |= ϕ, thenN |= ϕ.
Definition
![Page 96: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/96.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Existential homogeneity in free groups
Question : What can be said about existential homogeneity infree groups?Recall that:
Definition
Let M be a model and N a submodel of M. The model N is saidto be existentially closed (abbreviated e.c.) in M, if for anyexistential formula ϕ with parameters from N , if M |= ϕ, thenN |= ϕ.
Definition
Let F be a free group and let a = (a1, . . . , am) be a tuple fromF .
![Page 97: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/97.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Existential homogeneity in free groups
Question : What can be said about existential homogeneity infree groups?Recall that:
Definition
Let M be a model and N a submodel of M. The model N is saidto be existentially closed (abbreviated e.c.) in M, if for anyexistential formula ϕ with parameters from N , if M |= ϕ, thenN |= ϕ.
Definition
Let F be a free group and let a = (a1, . . . , am) be a tuple fromF .We say that a is a power of a primitive element if there existintegers p1, . . . , pm and a primitive element u such that ai = upi
for all i .
![Page 98: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/98.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
![Page 99: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/99.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Theorem 3
![Page 100: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/100.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Theorem 3
Let F be a nonabelian free group of finite rank.
![Page 101: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/101.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Theorem 3
Let F be a nonabelian free group of finite rank. Let a, b ∈ F n andP ⊆ F such that tpF
∃ (a|P) = tpF∃ (b|P).
![Page 102: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/102.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Theorem 3
Let F be a nonabelian free group of finite rank. Let a, b ∈ F n andP ⊆ F such that tpF
∃ (a|P) = tpF∃ (b|P). Then one of the following
cases holds:
![Page 103: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/103.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Theorem 3
Let F be a nonabelian free group of finite rank. Let a, b ∈ F n andP ⊆ F such that tpF
∃ (a|P) = tpF∃ (b|P). Then one of the following
cases holds:
the tuple a has the same existential type as a power of aprimitive element;
![Page 104: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/104.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Theorem 3
Let F be a nonabelian free group of finite rank. Let a, b ∈ F n andP ⊆ F such that tpF
∃ (a|P) = tpF∃ (b|P). Then one of the following
cases holds:
the tuple a has the same existential type as a power of aprimitive element;
there exists an e.c. subgroup E (a) (resp. E (b)) containing Pand a (resp. b) and an isomorphism σ : E (a) → E (b) fixingpointwise P and sending a to b.
![Page 105: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/105.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
![Page 106: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/106.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
We eliminate first parameters.
![Page 107: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/107.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
We eliminate first parameters. Recall that a free group isequationnally noetherian;
![Page 108: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/108.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
We eliminate first parameters. Recall that a free group isequationnally noetherian; that is any system of equations in finitelymany variable is equivalent to a finite subsystem.
![Page 109: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/109.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
We eliminate first parameters. Recall that a free group isequationnally noetherian; that is any system of equations in finitelymany variable is equivalent to a finite subsystem.
Lemma
![Page 110: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/110.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
We eliminate first parameters. Recall that a free group isequationnally noetherian; that is any system of equations in finitelymany variable is equivalent to a finite subsystem.
Lemma
Let G be a finitely generated equationally noetherian group.
![Page 111: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/111.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
We eliminate first parameters. Recall that a free group isequationnally noetherian; that is any system of equations in finitelymany variable is equivalent to a finite subsystem.
Lemma
Let G be a finitely generated equationally noetherian group. Let Pbe a subset of G .
![Page 112: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/112.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
We eliminate first parameters. Recall that a free group isequationnally noetherian; that is any system of equations in finitelymany variable is equivalent to a finite subsystem.
Lemma
Let G be a finitely generated equationally noetherian group. Let Pbe a subset of G . Then there exists a finite subset P0 ⊆ P suchthat for any endomorphism f of G , if f fixes pointwise P0 then ffixes pointwise P.
![Page 113: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/113.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
We eliminate first parameters. Recall that a free group isequationnally noetherian; that is any system of equations in finitelymany variable is equivalent to a finite subsystem.
Lemma
Let G be a finitely generated equationally noetherian group. Let Pbe a subset of G . Then there exists a finite subset P0 ⊆ P suchthat for any endomorphism f of G , if f fixes pointwise P0 then ffixes pointwise P.
Hence:
![Page 114: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/114.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
We eliminate first parameters. Recall that a free group isequationnally noetherian; that is any system of equations in finitelymany variable is equivalent to a finite subsystem.
Lemma
Let G be a finitely generated equationally noetherian group. Let Pbe a subset of G . Then there exists a finite subset P0 ⊆ P suchthat for any endomorphism f of G , if f fixes pointwise P0 then ffixes pointwise P.
Hence: ∃f ∈ Aut(F |P) s.t. f (a) = f (b) ⇔ ∃f ∈ Aut(F ) s.t.f (aP0) = f (bP0).
![Page 115: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/115.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
We eliminate first parameters. Recall that a free group isequationnally noetherian; that is any system of equations in finitelymany variable is equivalent to a finite subsystem.
Lemma
Let G be a finitely generated equationally noetherian group. Let Pbe a subset of G . Then there exists a finite subset P0 ⊆ P suchthat for any endomorphism f of G , if f fixes pointwise P0 then ffixes pointwise P.
Hence: ∃f ∈ Aut(F |P) s.t. f (a) = f (b) ⇔ ∃f ∈ Aut(F ) s.t.f (aP0) = f (bP0).Note that : If tp∃(a|P) = tp∃(b|P) then tp∃(a|P0) = tp∃(b|P0)and tp∃(aP0) = tp∃(bP0).
![Page 116: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/116.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
![Page 117: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/117.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
Definition
![Page 118: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/118.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
Definition
Let F1 and F2 be nonabelian free groups of finite rank and let a(resp. b) be a tuple from F1 (resp. F2).
![Page 119: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/119.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
Definition
Let F1 and F2 be nonabelian free groups of finite rank and let a(resp. b) be a tuple from F1 (resp. F2). We say that (a, b) isexistentially rigid, if there is no nontrivial free decompositionF1 = A ∗ B such that A contains a tuple c withtpF1
∃ (a) ⊆ tpA∃ (c) ⊆ tpF2
∃ (b).
![Page 120: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/120.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
![Page 121: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/121.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
Proposition (1)
![Page 122: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/122.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
Proposition (1)
Let F1 and F2 be nonabelian free groups of finite rank and let a(resp. b) be a tuple from F1 (resp. F2).
![Page 123: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/123.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
Proposition (1)
Let F1 and F2 be nonabelian free groups of finite rank and let a(resp. b) be a tuple from F1 (resp. F2). Suppose that (a, b) isexistentially rigid and let s be a basis of F1.
![Page 124: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/124.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
Proposition (1)
Let F1 and F2 be nonabelian free groups of finite rank and let a(resp. b) be a tuple from F1 (resp. F2). Suppose that (a, b) isexistentially rigid and let s be a basis of F1.Then there exists a quantifier-free formula ϕ(x , y), such thatF1 |= ϕ(a, s) and such that for any f ∈ Hom(F1|a,F2|b), ifF2 |= ϕ(b, f (s)) then f is an embedding.
![Page 125: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/125.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
![Page 126: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/126.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
Proposition (2)
![Page 127: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/127.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
Proposition (2)
Let F1 and F2 be nonabelian free groups of finite rank and a (resp.b) a tuple from F1 (resp. F2) such that tpF1
∃ (a) = tpF2
∃ (b).
![Page 128: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/128.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
Proposition (2)
Let F1 and F2 be nonabelian free groups of finite rank and a (resp.b) a tuple from F1 (resp. F2) such that tpF1
∃ (a) = tpF2
∃ (b).Suppose that (a, b) is existentially rigid.
![Page 129: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/129.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
Proposition (2)
Let F1 and F2 be nonabelian free groups of finite rank and a (resp.b) a tuple from F1 (resp. F2) such that tpF1
∃ (a) = tpF2
∃ (b).Suppose that (a, b) is existentially rigid.Then either rk(F1) = 2 and a is a power of a primitive element, orthere exists an embedding h ∈ Hom(F1|a,F2|b) such that h(F1) isan e.c. subgroup of F2.
![Page 130: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/130.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
![Page 131: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/131.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
Let F be a free group of finite rank and let a, b ∈ F n s.t.tpF
∃ (a) = tpF∃ (b).
![Page 132: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/132.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
Let F be a free group of finite rank and let a, b ∈ F n s.t.tpF
∃ (a) = tpF∃ (b).
Let C be the smallest free factor of F such that C contains c withtpF
∃ (a) = tpC∃ (c).
![Page 133: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/133.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
Let F be a free group of finite rank and let a, b ∈ F n s.t.tpF
∃ (a) = tpF∃ (b).
Let C be the smallest free factor of F such that C contains c withtpF
∃ (a) = tpC∃ (c).
Then (c , a) and (c , b) are existentially rigid.
![Page 134: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/134.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
Let F be a free group of finite rank and let a, b ∈ F n s.t.tpF
∃ (a) = tpF∃ (b).
Let C be the smallest free factor of F such that C contains c withtpF
∃ (a) = tpC∃ (c).
Then (c , a) and (c , b) are existentially rigid.By Propsition (2), either rk(C ) = 2 and c is a power of a primitiveelement or there exists an embedding h1 ∈ Hom(C |c ,F |a) (resp.h2 ∈ Hom(C |c ,F |b)) such that h1(C ) (resp. h2(C )) is an e.c.subgroup of F .
![Page 135: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/135.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
Let F be a free group of finite rank and let a, b ∈ F n s.t.tpF
∃ (a) = tpF∃ (b).
Let C be the smallest free factor of F such that C contains c withtpF
∃ (a) = tpC∃ (c).
Then (c , a) and (c , b) are existentially rigid.By Propsition (2), either rk(C ) = 2 and c is a power of a primitiveelement or there exists an embedding h1 ∈ Hom(C |c ,F |a) (resp.h2 ∈ Hom(C |c ,F |b)) such that h1(C ) (resp. h2(C )) is an e.c.subgroup of F .Suppose that c is not a power of a primitive element.
![Page 136: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/136.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Existential homogeneity & prime models
Sketch of proof of Theorem 3
Let F be a free group of finite rank and let a, b ∈ F n s.t.tpF
∃ (a) = tpF∃ (b).
Let C be the smallest free factor of F such that C contains c withtpF
∃ (a) = tpC∃ (c).
Then (c , a) and (c , b) are existentially rigid.By Propsition (2), either rk(C ) = 2 and c is a power of a primitiveelement or there exists an embedding h1 ∈ Hom(C |c ,F |a) (resp.h2 ∈ Hom(C |c ,F |b)) such that h1(C ) (resp. h2(C )) is an e.c.subgroup of F .Suppose that c is not a power of a primitive element. By settingE (a) = h1(C ) and E (b) = h2(C ), we have h2 ◦ h−1
1 : E (a) → E (b)is an isomorphism with h2 ◦ h−1
1 (a) = b.
![Page 137: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/137.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Homogeneity in free groups
![Page 138: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/138.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Homogeneity in free groups
Theorem 4
![Page 139: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/139.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Homogeneity in free groups
Theorem 4
Let F be a nonabelian free group of finite rank.
![Page 140: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/140.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Homogeneity in free groups
Theorem 4
Let F be a nonabelian free group of finite rank. For any tuplesa, b ∈ F n and for any subset P ⊆ F , if tpF (a|P) = tpF (b|P) thenthere exists an automorphism of F fixing pointwise P and sendinga to b.
![Page 141: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/141.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Homogeneity in free groups
Theorem 4
Let F be a nonabelian free group of finite rank. For any tuplesa, b ∈ F n and for any subset P ⊆ F , if tpF (a|P) = tpF (b|P) thenthere exists an automorphism of F fixing pointwise P and sendinga to b.
The above theorem is also proved by Perin and Sklinos.
![Page 142: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/142.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
![Page 143: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/143.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Z. Sela and O. Kharlampovich and A. Myasnikov show thatnonabelian free groups have the same elementary theory, and infact the following more explicit description.
![Page 144: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/144.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Z. Sela and O. Kharlampovich and A. Myasnikov show thatnonabelian free groups have the same elementary theory, and infact the following more explicit description.
Theorem 5
![Page 145: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/145.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Z. Sela and O. Kharlampovich and A. Myasnikov show thatnonabelian free groups have the same elementary theory, and infact the following more explicit description.
Theorem 5
A nonabelian free factor of a free group of finite rank is anelementary subgroup.
![Page 146: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/146.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
![Page 147: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/147.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
They show also the following quantifier-elimination result.
![Page 148: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/148.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
They show also the following quantifier-elimination result.
Theorem 6
![Page 149: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/149.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
They show also the following quantifier-elimination result.
Theorem 6
Let ϕ(x) be a formula.
![Page 150: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/150.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
They show also the following quantifier-elimination result.
Theorem 6
Let ϕ(x) be a formula. Then there exists a boolean combination of∃∀-forumulas φ(x), such that for any nonabelian free group F offinite rank, one has F |= ∀x(ϕ(x) ⇔ φ(x)).
![Page 151: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/151.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
They show also the following quantifier-elimination result.
Theorem 6
Let ϕ(x) be a formula. Then there exists a boolean combination of∃∀-forumulas φ(x), such that for any nonabelian free group F offinite rank, one has F |= ∀x(ϕ(x) ⇔ φ(x)).
We notice, in particular, that if a, b ∈ F n such thattpF
∃∀(a) = tpF∃∀(b), then tpF (a) = tpF (b).
![Page 152: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/152.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
![Page 153: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/153.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Theorem 7 (C. Perin, 2008)
![Page 154: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/154.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Theorem 7 (C. Perin, 2008)
An elementary subgroup of a free group of finite rank is a freefactor.
![Page 155: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/155.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Theorem 7 (C. Perin, 2008)
An elementary subgroup of a free group of finite rank is a freefactor.
Theorem 8 (A. Pillay, 2009)
![Page 156: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/156.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Theorem 7 (C. Perin, 2008)
An elementary subgroup of a free group of finite rank is a freefactor.
Theorem 8 (A. Pillay, 2009)
Let F be a nonabelian free group of finite rank and u, v ∈ F suchthat tpF (u) = tpF (v).
![Page 157: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/157.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Theorem 7 (C. Perin, 2008)
An elementary subgroup of a free group of finite rank is a freefactor.
Theorem 8 (A. Pillay, 2009)
Let F be a nonabelian free group of finite rank and u, v ∈ F suchthat tpF (u) = tpF (v). If u is primitive, then v is primitive.
![Page 158: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/158.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
![Page 159: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/159.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Definition
![Page 160: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/160.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Definition
Let F be nonabelian free group of finite rank and let a be a tupleof F .
![Page 161: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/161.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Definition
Let F be nonabelian free group of finite rank and let a be a tupleof F . We say that a is rigid if there is no nontrivial freedecomposition F = A ∗ B such that A contains a tuple c withtpF1
∃∀(a) = tpA∃∀(c).
![Page 162: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/162.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
![Page 163: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/163.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Proposition (3)
![Page 164: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/164.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Proposition (3)
Let F1 and F2 be nonabelian free groups of finite rank and let a(resp. b) be a tuple from F1 (resp. F2) such thattpF1
∃∀(a) = tpF2
∃∀(b).
![Page 165: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/165.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Proposition (3)
Let F1 and F2 be nonabelian free groups of finite rank and let a(resp. b) be a tuple from F1 (resp. F2) such thattpF1
∃∀(a) = tpF2
∃∀(b). Suppose that a is rigid.
![Page 166: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/166.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Proposition (3)
Let F1 and F2 be nonabelian free groups of finite rank and let a(resp. b) be a tuple from F1 (resp. F2) such thattpF1
∃∀(a) = tpF2
∃∀(b). Suppose that a is rigid.Then either rk(F1) = 2 and a is a power of a primitive element, orthere exists an embedding h ∈ Hom(F1|a,F2|b) such thath(F1) �∃∀ F2.
![Page 167: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/167.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
![Page 168: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/168.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Let F be a free group of finite rank and let a, b ∈ F n s.t.tpF (a) = tpF (b).
![Page 169: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/169.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Let F be a free group of finite rank and let a, b ∈ F n s.t.tpF (a) = tpF (b).Let C be the smallest free factor of F such that C contains c withtpF (a) = tpC (c).
![Page 170: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/170.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Let F be a free group of finite rank and let a, b ∈ F n s.t.tpF (a) = tpF (b).Let C be the smallest free factor of F such that C contains c withtpF (a) = tpC (c).Then c is rigid.
![Page 171: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/171.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Let F be a free group of finite rank and let a, b ∈ F n s.t.tpF (a) = tpF (b).Let C be the smallest free factor of F such that C contains c withtpF (a) = tpC (c).Then c is rigid.By Propsition (3), either rk(C ) = 2 and c is a power of a primitiveelement or there exists an embedding h1 ∈ Hom(C |a,F |b) (resp.h2 ∈ Hom(C |a,F |b)) such that h1(C ) �∃∀ F (resp. h2(C ) �∃∀ F ).
![Page 172: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/172.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Let F be a free group of finite rank and let a, b ∈ F n s.t.tpF (a) = tpF (b).Let C be the smallest free factor of F such that C contains c withtpF (a) = tpC (c).Then c is rigid.By Propsition (3), either rk(C ) = 2 and c is a power of a primitiveelement or there exists an embedding h1 ∈ Hom(C |a,F |b) (resp.h2 ∈ Hom(C |a,F |b)) such that h1(C ) �∃∀ F (resp. h2(C ) �∃∀ F ).
If c is a power of a primitive element then the result follows fromTheorem 8.
![Page 173: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/173.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
![Page 174: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/174.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Suppose that c is not a power of a primitive element.
![Page 175: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/175.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Suppose that c is not a power of a primitive element. SetE (a) = h1(C ) and E (b) = h2(C ).
![Page 176: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/176.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Suppose that c is not a power of a primitive element. SetE (a) = h1(C ) and E (b) = h2(C ).We have h2 ◦ h−1
1 : E (a) → E (b) is an isomorphism withh2 ◦ h−1
1 (a) = b.
![Page 177: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/177.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Suppose that c is not a power of a primitive element. SetE (a) = h1(C ) and E (b) = h2(C ).We have h2 ◦ h−1
1 : E (a) → E (b) is an isomorphism withh2 ◦ h−1
1 (a) = b.Since E (a) �∃∀ F (resp. E (b) �∃∀ F ) we get by Theorem 7 thatE (a) � F (resp. E (b) � F ).
![Page 178: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/178.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Suppose that c is not a power of a primitive element. SetE (a) = h1(C ) and E (b) = h2(C ).We have h2 ◦ h−1
1 : E (a) → E (b) is an isomorphism withh2 ◦ h−1
1 (a) = b.Since E (a) �∃∀ F (resp. E (b) �∃∀ F ) we get by Theorem 7 thatE (a) � F (resp. E (b) � F ).By Theorem 7, E (a) (resp. E (b)) is a free factor of F .
![Page 179: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/179.jpg)
Homogeneity and algebraic closure in free groups
Homogeneity & prime models
Homogeneity
Sketch of proof of Theorem 4
Suppose that c is not a power of a primitive element. SetE (a) = h1(C ) and E (b) = h2(C ).We have h2 ◦ h−1
1 : E (a) → E (b) is an isomorphism withh2 ◦ h−1
1 (a) = b.Since E (a) �∃∀ F (resp. E (b) �∃∀ F ) we get by Theorem 7 thatE (a) � F (resp. E (b) � F ).By Theorem 7, E (a) (resp. E (b)) is a free factor of F .Therefore h2 ◦ h−1
1 can be extended to an isomorphism of F .
![Page 180: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/180.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Contents
1 Homogeneity & prime modelsDefinitionsExistential homogeneity & prime modelsHomogeneity
2 Algebraic & definable closureDefinitionsConstructibility over the algebraic closureA counterexample
![Page 181: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/181.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Definitions
Definitions
![Page 182: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/182.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Definitions
Definitions
This part is a joint work with D. Vallino.
![Page 183: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/183.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Definitions
Definitions
This part is a joint work with D. Vallino.Recall that:
![Page 184: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/184.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Definitions
Definitions
This part is a joint work with D. Vallino.Recall that:
Definition
Let G be a group and A a subset of G .
![Page 185: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/185.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Definitions
Definitions
This part is a joint work with D. Vallino.Recall that:
Definition
Let G be a group and A a subset of G .
The algebraic closure of A, denoted aclG (A), is the set ofelements g ∈ G such that there exists a formula φ(x) withparameters from A such that G |= φ(g) and φ(G ) is finite.
![Page 186: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/186.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Definitions
Definitions
This part is a joint work with D. Vallino.Recall that:
Definition
Let G be a group and A a subset of G .
The algebraic closure of A, denoted aclG (A), is the set ofelements g ∈ G such that there exists a formula φ(x) withparameters from A such that G |= φ(g) and φ(G ) is finite.
The definable closure of A, denoted dclG (A), is the set ofelements g ∈ G such that there exists a formula φ(x) withparameters from A such that G |= φ(g) and φ(G ) is asingleton.
![Page 187: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/187.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
![Page 188: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/188.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Question (Z. Sela, 2008):
![Page 189: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/189.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Question (Z. Sela, 2008): Is it true that acl(A) = dcl(A) in freegroups?
![Page 190: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/190.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Question (Z. Sela, 2008): Is it true that acl(A) = dcl(A) in freegroups?Remarks.
![Page 191: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/191.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Question (Z. Sela, 2008): Is it true that acl(A) = dcl(A) in freegroups?Remarks. Let Γ be a torsion-free hyperbolic group and A ⊆ Γ.
![Page 192: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/192.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Question (Z. Sela, 2008): Is it true that acl(A) = dcl(A) in freegroups?Remarks. Let Γ be a torsion-free hyperbolic group and A ⊆ Γ.
acl(A) = acl(〈A〉) and dcl(A) = dcl(〈A〉).
![Page 193: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/193.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Question (Z. Sela, 2008): Is it true that acl(A) = dcl(A) in freegroups?Remarks. Let Γ be a torsion-free hyperbolic group and A ⊆ Γ.
acl(A) = acl(〈A〉) and dcl(A) = dcl(〈A〉). Hence, we mayassume that A is a subgroup.
![Page 194: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/194.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Question (Z. Sela, 2008): Is it true that acl(A) = dcl(A) in freegroups?Remarks. Let Γ be a torsion-free hyperbolic group and A ⊆ Γ.
acl(A) = acl(〈A〉) and dcl(A) = dcl(〈A〉). Hence, we mayassume that A is a subgroup.
If Γ = Γ1 ∗ Γ2 and A ≤ Γ1 then acl(A) ≤ Γ1.
![Page 195: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/195.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Question (Z. Sela, 2008): Is it true that acl(A) = dcl(A) in freegroups?Remarks. Let Γ be a torsion-free hyperbolic group and A ⊆ Γ.
acl(A) = acl(〈A〉) and dcl(A) = dcl(〈A〉). Hence, we mayassume that A is a subgroup.
If Γ = Γ1 ∗ Γ2 and A ≤ Γ1 then acl(A) ≤ Γ1. Similarly fordcl(A).
![Page 196: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/196.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Question (Z. Sela, 2008): Is it true that acl(A) = dcl(A) in freegroups?Remarks. Let Γ be a torsion-free hyperbolic group and A ⊆ Γ.
acl(A) = acl(〈A〉) and dcl(A) = dcl(〈A〉). Hence, we mayassume that A is a subgroup.
If Γ = Γ1 ∗ Γ2 and A ≤ Γ1 then acl(A) ≤ Γ1. Similarly fordcl(A). Hence, we may assume that Γ is freelyA-indecomposable.
![Page 197: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/197.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Question (Z. Sela, 2008): Is it true that acl(A) = dcl(A) in freegroups?Remarks. Let Γ be a torsion-free hyperbolic group and A ⊆ Γ.
acl(A) = acl(〈A〉) and dcl(A) = dcl(〈A〉). Hence, we mayassume that A is a subgroup.
If Γ = Γ1 ∗ Γ2 and A ≤ Γ1 then acl(A) ≤ Γ1. Similarly fordcl(A). Hence, we may assume that Γ is freelyA-indecomposable.
If A is abelian then acl(A) = dcl(A) = CΓ(A).
![Page 198: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/198.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Question (Z. Sela, 2008): Is it true that acl(A) = dcl(A) in freegroups?Remarks. Let Γ be a torsion-free hyperbolic group and A ⊆ Γ.
acl(A) = acl(〈A〉) and dcl(A) = dcl(〈A〉). Hence, we mayassume that A is a subgroup.
If Γ = Γ1 ∗ Γ2 and A ≤ Γ1 then acl(A) ≤ Γ1. Similarly fordcl(A). Hence, we may assume that Γ is freelyA-indecomposable.
If A is abelian then acl(A) = dcl(A) = CΓ(A). Hence , wemay assume that A is nonabelian.
![Page 199: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/199.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
![Page 200: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/200.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Theorem 9
![Page 201: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/201.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Theorem 9
Let Γ be a torsion-free hyperbolic group and A ≤ Γ where A isnonabelian.
![Page 202: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/202.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Theorem 9
Let Γ be a torsion-free hyperbolic group and A ≤ Γ where A isnonabelian.Then Γ can be constructed from acl(A) by a finite sequence of freeproducts and HNN-extensions along cyclic subgroups.
![Page 203: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/203.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Theorem 9
Let Γ be a torsion-free hyperbolic group and A ≤ Γ where A isnonabelian.Then Γ can be constructed from acl(A) by a finite sequence of freeproducts and HNN-extensions along cyclic subgroups.In particular, acl(A) is finitely generated, quasiconvex andhyperbolic.
![Page 204: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/204.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Acl and the JSJ-decomposition
![Page 205: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/205.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Acl and the JSJ-decomposition
Theorem 10
![Page 206: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/206.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Acl and the JSJ-decomposition
Theorem 10
Let F be a nonabelian free group of finite rank and let A ≤ F be anonabelian subgroup.
![Page 207: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/207.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Acl and the JSJ-decomposition
Theorem 10
Let F be a nonabelian free group of finite rank and let A ≤ F be anonabelian subgroup. Then:
![Page 208: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/208.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Acl and the JSJ-decomposition
Theorem 10
Let F be a nonabelian free group of finite rank and let A ≤ F be anonabelian subgroup. Then:
dcl(A) is a free factor of acl(A).
![Page 209: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/209.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Acl and the JSJ-decomposition
Theorem 10
Let F be a nonabelian free group of finite rank and let A ≤ F be anonabelian subgroup. Then:
dcl(A) is a free factor of acl(A). In particular, if rk(F ) = 2then acl(A) = dcl(A).
![Page 210: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/210.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
Constructibility over the algebraic closure
Acl and the JSJ-decomposition
Theorem 10
Let F be a nonabelian free group of finite rank and let A ≤ F be anonabelian subgroup. Then:
dcl(A) is a free factor of acl(A). In particular, if rk(F ) = 2then acl(A) = dcl(A).
acl(A) is exactly the vertex group containing A in the cyclicJSJ-decomposition of F relative to A.
![Page 211: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/211.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
A counterexample
A counterexample
![Page 212: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/212.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
A counterexample
A counterexample
Theorem 11
![Page 213: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/213.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
A counterexample
A counterexample
Theorem 11
Let A0 be a finite set (possibly empty)
![Page 214: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/214.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
A counterexample
A counterexample
Theorem 11
Let A0 be a finite set (possibly empty) and
A = 〈A0, a, b, u|〉, H = A ∗ 〈y |〉,
![Page 215: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/215.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
A counterexample
A counterexample
Theorem 11
Let A0 be a finite set (possibly empty) and
A = 〈A0, a, b, u|〉, H = A ∗ 〈y |〉,
v = aybyay−1by−1,
![Page 216: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/216.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
A counterexample
A counterexample
Theorem 11
Let A0 be a finite set (possibly empty) and
A = 〈A0, a, b, u|〉, H = A ∗ 〈y |〉,
v = aybyay−1by−1, F = 〈H, t|ut = v〉.
![Page 217: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/217.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
A counterexample
A counterexample
Theorem 11
Let A0 be a finite set (possibly empty) and
A = 〈A0, a, b, u|〉, H = A ∗ 〈y |〉,
v = aybyay−1by−1, F = 〈H, t|ut = v〉.
Then F is a free group of rank |A0| + 4 and:
![Page 218: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/218.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
A counterexample
A counterexample
Theorem 11
Let A0 be a finite set (possibly empty) and
A = 〈A0, a, b, u|〉, H = A ∗ 〈y |〉,
v = aybyay−1by−1, F = 〈H, t|ut = v〉.
Then F is a free group of rank |A0| + 4 and:
If f ∈ Hom(F |A) then f ∈ Aut(F |A), and if f|H 6= 1 thenf (y) = y−1.
![Page 219: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/219.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
A counterexample
A counterexample
Theorem 11
Let A0 be a finite set (possibly empty) and
A = 〈A0, a, b, u|〉, H = A ∗ 〈y |〉,
v = aybyay−1by−1, F = 〈H, t|ut = v〉.
Then F is a free group of rank |A0| + 4 and:
If f ∈ Hom(F |A) then f ∈ Aut(F |A), and if f|H 6= 1 thenf (y) = y−1.
acl(A) = acl∃(A) = H.
![Page 220: Homogeneity and algebraic closure in free groupsmath.univ-lyon1.fr/~ould/Fiches-papiers/expo-crm-montreal.pdf · Mapping class groups of closed surfaces (N.V. Ivanov, J.D. McCarthy,](https://reader033.fdocuments.us/reader033/viewer/2022050123/5f5329cda44adc65721efe87/html5/thumbnails/220.jpg)
Homogeneity and algebraic closure in free groups
Algebraic & definable closure
A counterexample
A counterexample
Theorem 11
Let A0 be a finite set (possibly empty) and
A = 〈A0, a, b, u|〉, H = A ∗ 〈y |〉,
v = aybyay−1by−1, F = 〈H, t|ut = v〉.
Then F is a free group of rank |A0| + 4 and:
If f ∈ Hom(F |A) then f ∈ Aut(F |A), and if f|H 6= 1 thenf (y) = y−1.
acl(A) = acl∃(A) = H.
dcl(A) = dcl∃(A) = A.