BIBFRAME Work

Title

Towers and the first-order theories of hyperbolic groups

Type

• Monograph
• Text

Contribution

• Sklinos, Rizos (author )
• Guirardel, V. (Vincent), 1973- (author )
• Levitt, Gilbert, 1955- (author )

Subject

• Group theory and generalizations -- Special aspects of infinite or finite groups -- Hyperbolic groups and nonpositively curved groups (MSC)
• Mathematical logic and foundations -- Model theory -- Model-theoretic algebra (MSC)
• Model theory (LCSH)
• Hyperbolic groups (LCSH)

Language

• eng

Illustrative Content

• illustrations

Classification

• DDC: 512/.2 (Assigner: dlc)
• LCC: QA171 .G85 2024 (Assigner: dlc) (Status: uba)
• 20F67
• 03C60

Content

• text

Supplementary content

• bibliography

Summary

"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 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 counter examples, 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 Fraısse's method, we associate to every torsion-free hyperbolic group H a unique homogeneous countable groupMin 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" Provided by publisher.

Table of Contents

Preliminaries -- Elementary equivalence and elementarily embedded subgroups -- Surfaces -- Splittings and trees -- Grushko decompositions.

Authorized Access Point

Guirardel, V. (Vincent), 1973-. Towers and the first-order theories of hyperbolic groups

Admin Metadata

• Date: 2026-09-11T20:11:18.663968+00:00
• Agent: cbc
• Status: c
• Derived from: 23762544

Admin Metadata

• Date: 2026-09-11
• Cataloger id: Unknown
• Description level: bibframe-2-6-0
• Description language: eng
• Description authentication: pcc