Acylindrical hyperbolicity and existential closedness

André, Simon

doi:10.48550/arxiv.2005.07220

Cited by 1 publication

(6 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(3) α j is a modular automorphism of type (3), that is a natural extension of an automorphism α v of a vertex group L v of R L of Seifert-type, as described in Definition 3. 23.…”

Section: Approximations Of Limit Groupsmentioning

confidence: 99%

“…Proposition 6. 3. Let G be an acylindrically hyperbolic group, and let a be a tuple of elements of G. Fix a presentation a | R(a) = 1 for the subgroup of G generated by a.…”

Section: It Follows Thatmentioning

confidence: 99%

“…Before proving this claim, let us explain how to complete the proof of Proposition 6. 3. First, note that if q i+1 (A U i ) is equal to A U i+1 , then if one shortens the restriction of θ i+1 n to A U i+1 , one automatically shortens the restriction of ρ i n to A U i , which is not possible by definition of ρ i n .…”

Section: It Follows Thatmentioning

confidence: 99%

“…In this section, we prove Theorem 1. 3. First, recall that this theorem says that every acylindrically hyperbolic group G is ∃∀∃-embedded into the HNN extensions…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

“…Question 10. 3. Is acylindrical hyperbolicity preserved under elementary equivalence among finitely generated groups?…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

See 4 more Smart Citations

Formal solutions and the first-order theory of acylindrically hyperbolic groups

André,

Fruchter

2020

Preprint

Self Cite

View full text Add to dashboard Cite

We generalise Merzlyakov's theorem about the first-order theory of nonabelian free groups to all acylindrically hyperbolic groups. As a corollary, we deduce that if G is an acylindrically hyperbolic group and E(G) denotes the unique maximal finite normal subgroup of G, then G and the HNN extension G * E(G) , which is simply the free product G * Z when E(G) is trivial, have the same ∀∃-theory. As a consequence, we prove the following conjecture, formulated by Casals-Ruiz, Garreta and de la Nuez González: acylindrically hyperbolic groups have trivial positive theory. In particular, one recovers a result proved by Bestvina, Bromberg and Fujiwara, stating that, with only the obvious exceptions, verbal subgroups of acylindrically hyperbolic groups have infinite width.(2) Is G elementarily embedded into G * E(G) ?As mentioned before, Sela proved that the answer to both of these questions is 'Yes' under the stronger assumption that G is a torsion-free non-elementary hyperbolic group or a non-trivial and non-dihedral free product. In all other cases, the answer is not known.Moreover, let us note that we do not know of any example of a finitely generated group G that is not acylindrically hyperbolic and that has the same first-order theory, or even the same ∀∃-theory, as G * Z. The question of the existence of such a group is closely related to that of the preservation of acylindrical hyperbolicity under elementary equivalence among finitely generated groups (see Section 10 for further comments). It is worth mentioning the following corollary of Theorem 1.3 (see Proposition 10.5).Corollary 1.7. Let G be an acylindrically hyperbolic group, and let H be a group that admits a non-trivial splitting over a virtually abelian group. Suppose that G and H are elementarily equivalent (or simply that they have the same ∃∀∃-theory). Then the group H is acylindrically hyperbolic.Remark 1.8. As a consequence, if there exists a group G that is not acylindrically hyperbolic and such that G and G * Z are elementarily equivalent, then all non-trivial splittings of G (if they exist) have sufficiently complicated edge groups. For instance, if G is a generalized Baumslag-Solitar group, then G and G * Z are not elementarily equivalent.

show abstract

“…(3) α j is a modular automorphism of type (3), that is a natural extension of an automorphism α v of a vertex group L v of R L of Seifert-type, as described in Definition 3. 23.…”

Section: Approximations Of Limit Groupsmentioning

confidence: 99%

“…Proposition 6. 3. Let G be an acylindrically hyperbolic group, and let a be a tuple of elements of G. Fix a presentation a | R(a) = 1 for the subgroup of G generated by a.…”

Section: It Follows Thatmentioning

confidence: 99%

Section: It Follows Thatmentioning

confidence: 99%

“…In this section, we prove Theorem 1. 3. First, recall that this theorem says that every acylindrically hyperbolic group G is ∃∀∃-embedded into the HNN extensions…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

“…Question 10. 3. Is acylindrical hyperbolicity preserved under elementary equivalence among finitely generated groups?…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

See 3 more Smart Citations

Formal solutions and the first-order theory of acylindrically hyperbolic groups

André,

Fruchter

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Acylindrical hyperbolicity and existential closedness

Cited by 1 publication

References 13 publications

Formal solutions and the first-order theory of acylindrically hyperbolic groups

Formal solutions and the first-order theory of acylindrically hyperbolic groups

Contact Info

Product

Resources

About