Homogeneity and Prime Models in Torsion-Free Hyperbolic Groups

Houcine, Abderezak Ould

doi:10.1142/s179374421100028x

Cited by 38 publications

(38 citation statements)

References 26 publications

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“…every type can in fact be defined by a single formula (see [PS16], [OH11]). This enables us on the one hand to transfer a sequence witnessing forking from a big model to F, but more importantly it gives us a natural candidate for a formula witnessing forking: by homogeneity (see [PS12] and [OH11]), types in F correspond to orbits under the automorphism group, so in fact the orbit of a tuple under Aut A (F) is definable. Geometrically, under the assumption that A is not contained in any proper free factor, we have a very good understanding of Aut A (F): the canonical JSJ decomposition of F relative to A enables one to describe up to finite index the automorphisms fixing A (one understands the modular automorpshism group Mod A (F)).…”

Section: Introductionmentioning

confidence: 99%

Forking and JSJ decompositions in the free group

Perin¹,

Sklinos²

2016

J. Eur. Math. Soc.

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Section: Introductionmentioning

confidence: 99%

Forking and JSJ decompositions in the free group

Perin¹,

Sklinos²

2016

J. Eur. Math. Soc.

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“…The same result is true for A Γ without the center because the standard generating set is definable. Notice that non-abelian free groups are also homogeneous [9], [7]. Definition 2.…”

Section: Definability Of a Submonoidmentioning

confidence: 99%

Bi-interpretability of some monoids with the arithmetic and applications

Kharlampovich

López

2019

Semigroup Forum

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We will prove bi-interpretability of the arithmetic N = N, +, ·, 0, 1 and the weak second order theory of N with the free monoid MX of finite rank greater than 1 and with a non-trivial partially commutative monoid with trivial center. This bi-interpretability implies that finitely generated submonoids of these monoids are definable. Moreover, any recursively enumerable language in the alphabet X is definable in MX . Primitive elements, and, therefore, free bases are definable in the free monoid. It has the so-called QFA property, namely there is a sentence φ such that every finitely generated monoid satisfying φ is isomorphic to MX . The same is true for a partially commutative monoid without center. We also prove that there is no quantifier elimination in the theory of any structure that is bi-interpretable with N to any boolean combination of formulas from Πn or Σn. * Hunter College, CUNY † Graduate Center, CUNYIt follows from Quine's paper [10], Section 4, that for a free monoid of rank n ≥ 2 and generating set X = {x 1 , . . . , x n }, the arithmetic N, +, ·, ↑, 0, 1 is bi-interpretable in M X with parameters X, where x ↑ y means x y . Since the predicate z = x y is computable and therefore definable in terms of addition and multiplication (see, for example, [5]) it can be removed from the signature. Quine was working with the structure of concatenation with parameters, C = C, ⌢ , where C is the set of all finite strings in a finite alphabet and ⌢ is the concatenation operation. This structure C and the free semigroup are equivalent structures. Quine did not state the monoid version of his results but the presence of the identity doesn't make a difference and his results are also valid for M X .We will show that N and the weak second order theory of N are bi-interpretable with M X (with parameters X) using the technique that allows also to prove this for a non-trivial free partially commutative monoid A Γ with trivial center. Here we say that the weak second order theory of a structure B is interpretable in A if the first-order structure S(B, N) (see Section 2.1 for precise definition) which has the same expressive power, is interpretable in A.This bi-interpretability has interesting applications. For example, Theorem 4 states that for any k ∈ N, there is a formula ψ(y, y 1 , . . . , y k , X) such that ψ(g, g 1 , . . . , g k , X) holds in M X if and only if g belongs to the submonoid generated by g 1 , . . . , g k . In other words, finitely generated submonoids of M X are definable. In contrast to this, it was proved in [4] and later in [8] that proper subgroups of a free group are not definable (except cyclic subgroups when the language contains constants). This was a solution of an old problem posed by Malcev. Primitive elements, and, therefore, free bases are not definable in a free group of rank greater than 2 [4], but they are easily definable in the free monoid. This implies that the free monoid is homogeneous (two tuples realize the same types if and only if they are automorphically equivalent). We wil...

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“…It is shown in [OH11] and [PS10] that nonabelian free groups of finite rank are homogeneous. In the sequel we need the following theorem proved in [OH11].…”

Section: Proofmentioning

confidence: 99%

Algebraic and definable closure in free groups

Houcine

Vallino

2016

Ann. inst. Fourier

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We study algebraic closure and its relation with definable closure in free groups and more generally in torsion-free hyperbolic groups. Given a torsion-free hyperbolic group Γ and a nonabelian subgroup A of Γ, we describe Γ as a constructible group from the algebraic closure of A along cyclic subgroups. In particular, it follows that the algebraic closure of A is finitely generated, quasiconvex and hyperbolic.Suppose that Γ is free. Then the definable closure of A is a free factor of the algebraic closure of A and the rank of these groups is bounded by that of Γ. We prove that the algebraic closure of A coincides with the vertex group containing A in the generalized malnormal cyclic JSJ-decomposition of Γ relative to A. If the rank of Γ is bigger than 4, then Γ has a subgroup A such that the definable closure of A is a proper subgroup of the algebraic closure of A. This answers a question of Sela.

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Homogeneity and Prime Models in Torsion-Free Hyperbolic Groups

Cited by 38 publications

References 26 publications

Forking and JSJ decompositions in the free group

Forking and JSJ decompositions in the free group

Bi-interpretability of some monoids with the arithmetic and applications

Algebraic and definable closure in free groups

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