Hyperbolicity and cubulability are preserved under elementary equivalence

André, Simon

doi:10.2140/gt.2020.24.1075

Cited by 4 publications

(2 citation statements)

References 28 publications

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“…In [1], the first author proved that the property of being a hyperbolic group is preserved under elementary equivalence among finitely generated groups (this result was proved by Sela in [37] for torsion‐free groups). Since acylindrically hyperbolic groups are not supposed to be finitely generated, Question 10.3 makes sense without assuming finite generation; however, the answer to this question is negative in general, even among countable groups.…”

Section: Questions and Commentsmentioning

confidence: 99%

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

André

Fruchter

2022

Journal of London Math Soc

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We generalise Merzlyakov's theorem about the first‐order theory of non‐abelian free groups to all acylindrically hyperbolic groups. As a corollary, we deduce that if G$G$ is an acylindrically hyperbolic group and Efalse(Gfalse)$E(G)$ denotes the unique maximal finite normal subgroup of G$G$, then G$G$ and the HNN extension G*̇Efalse(Gfalse)$G\dot{\ast }_{E(G)}$, which is simply the free product G*double-struckZ$G\ast \mathbb {Z}$ when Efalse(Gfalse)$E(G)$ is trivial, have the same ∀∃$\forall \exists$‐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.

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Section: Questions and Commentsmentioning

confidence: 99%

“…(1) 𝛼 𝑗 is a modular automorphism of 𝐿 of type (1), that is conjugation by some element g ∈ 𝐿. Let g ′ ∈ 𝐴 be such that 𝜃 ∞ (g ′ ) = g and set 𝛽 𝑗 to be conjugation by g ′ .…”

Section: The Shortening Argumentmentioning

confidence: 99%

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

André

Fruchter

2022

Journal of London Math Soc

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Homogeneity in virtually free groups

André

2022

Isr. J. Math.

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Towers and the First-order Theories of Hyperbolic Groups

Guirardel,

Levitt,

Sklinos

2024

Memoirs of the AMS

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This paper is devoted to the first-order theories of torsion-free hyperbolic groups. One of its purposes is to review some results and to provide precise and correct statements and definitions, as well as some proofs and new results. A key concept is that of a tower (Sela) or NTQ system (Kharlampovich-Myasnikov). We discuss them thoroughly. We state and prove a new general theorem which unifies several results in the literature: elementarily equivalent torsion-free hyperbolic groups have isomorphic cores (Sela); if H H is elementarily embedded in a torsion-free hyperbolic group G G , then G G is a tower over H H relative to H H (Perin); free groups (Perin-Sklinos, Ould-Houcine), and more generally free products of prototypes and free groups, are homogeneous. The converse to Sela and Perin’s results just mentioned is true. This follows from the solution to Tarski’s problem on elementary equivalence of free groups, due independently to Sela and Kharlampovich-Myasnikov, which we treat as a black box throughout the paper. We present many examples and counterexamples, and we prove some new model-theoretic results. We characterize prime models among torsion-free hyperbolic groups, and minimal models among elementarily free groups. Using Fraïssé’s method, we associate to every torsion-free hyperbolic group H H a unique homogeneous countable group M {\mathcal {M}} in which any hyperbolic group H ′ H’ elementarily equivalent to H H has an elementary embedding. In an appendix we give a complete proof of the fact, due to Sela, that towers over a torsion-free hyperbolic group H H are H H -limit groups.

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Hyperbolicity and cubulability are preserved under elementary equivalence

Cited by 4 publications

References 28 publications

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

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

Homogeneity in virtually free groups

Towers and the First-order Theories of Hyperbolic Groups

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