Elementary equivalence vs. commensurability for hyperbolic groups

Guirardel, Vincent; Levitt, Gilbert; Sklinos, Rizos

doi:10.1090/tran/7392

Cited by 3 publications

(20 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If Γ is simple and B 1 Z, we say that G is a parachute (parachutes are studied in Propositions 6.7 and 6.8 of [GLS19] and in Subsection 10.6).…”

Section: Centered Splittingsmentioning

confidence: 99%

“…The first two additional conditions ensure that p carries meaningful information; in particular, it is not the identity and does not factor through an abelian group. Non-degenerate boundary-preserving maps were called local preretractions in [GLS19].…”

Section: Boundary-preserving Maps Pinched Quotientmentioning

confidence: 99%

“…Remark 4.25. Recall (Corollary 3.5 of [GLS19]) that, if a centered splitting Γ is retractable, then g ≥ n 1 , with g the genus of Σ and n 1 the number of bottom vertices of Γ having valence 1.…”

Section: There Existmentioning

confidence: 99%

“…Since H is a retract of G, it suffices to find a homomorphism from G onto Z/2Z vanishing on H. This is easy to do if Σ is not planar or if Γ is not a tree of groups. In the remaining case (Σ is planar and Γ is a tree), Γ cannot be retractable [GLS19]: a retractable Γ with Γ planar has no terminal vertex (see Remark 4.25).…”

Section: There Existmentioning

confidence: 99%

“…One may require the existence of a retraction ρ : G → B 1 such that π 1 (Σ) has non-abelian image (in the example, ρ sends x to a and y to b if g = [a, b]). One may also require the existence of a map r from π 1 (Σ) to G (such as the restriction of ρ to π 1 (Σ)) which is a non-degenerate boundary-preserving map (called local preretraction in [GLS19]): it must agree with a conjugation on each boundary subgroup π 1 (C i ), have non-abelian image, not be an isomorphism between π 1 (Σ) and some conjugate.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Towers and the first-order theory of hyperbolic groups

Guirardel¹,

Levitt²,

Sklinos³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

This paper is devoted to the first-order theory 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 is elementarily embedded in a torsion-free hyperbolic group G, then G is a tower over H relative to 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 a unique homogeneous countable group M in which any hyperbolic group H elementarily equivalent to 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 are H-limit groups.

show abstract

“…If Γ is simple and B 1 Z, we say that G is a parachute (parachutes are studied in Propositions 6.7 and 6.8 of [GLS19] and in Subsection 10.6).…”

Section: Centered Splittingsmentioning

confidence: 99%

Section: Boundary-preserving Maps Pinched Quotientmentioning

confidence: 99%

Section: There Existmentioning

confidence: 99%

Section: There Existmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Towers and the first-order theory of hyperbolic groups

Guirardel¹,

Levitt²,

Sklinos³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Towers and the First-order Theories of Hyperbolic Groups

Guirardel,

Levitt,

Sklinos

2024

Memoirs of the AMS

View full text Add to dashboard Cite

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.

show abstract

Towers and elementary embeddings in toral relatively hyperbolic groups

Perez¹

2020

Preprint

View full text Add to dashboard Cite

In a remarkable series of papers, Zlil Sela classified the first-order theories of free groups and torsion-free hyperbolic groups using geometric structures he called towers. It was later proved by Chlo Perin that if H is an elementarily embedded subgroup (or elementary submodel) of a torsion-free hyperbolic group G, then G is a tower over H. We prove a generalization of Perin's result to toral relatively hyperbolic groups using JSJ and shortening techniques.

show abstract

Elementary equivalence vs. commensurability for hyperbolic groups

Cited by 3 publications

References 19 publications

Towers and the first-order theory of hyperbolic groups

Towers and the first-order theory of hyperbolic groups

Towers and the First-order Theories of Hyperbolic Groups

Towers and elementary embeddings in toral relatively hyperbolic groups

Contact Info

Product

Resources

About