Acylindrically hyperbolic groups

2015/05/22 by Denis Osin · 1 citation
Mathematics · #Geometric and Algebraic Topology #Advanced Algebra and Geometry #Homotopy and Cohomology in Algebraic Topology #Algorithm #Semantics (computer science) #Annotation #Mathematics #Class (philosophy) #Artificial intelligence #Computer science

paper · pdf · doi:10.1090/tran/6343

openalex publication_date 2015/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/18

Abstract

We say that a group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <italic>acylindrically hyperbolic</italic> if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that the class of acylindrically hyperbolic groups coincides with many other classes studied in the literature, e.g., the class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript g e o m"> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mi>e</mml:mi> <mml:mi>o</mml:mi> <mml:mi>m</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">C_geom</mml:annotation> </mml:semantics> </mml:math> </inline-formula> introduced by Hamenstädt, the class of groups admitting a non-elementary weakly properly discontinuous action on a hyperbolic space in the sense of Bestvina and Fujiwara, and the class of groups with hyperbolically embedded subgroups studied by Dahmani, Guirardel, and the author. We also record some basic results about acylindrically hyperbolic groups for future use.

Citations

• Géométrie et théorie des groupes
• Small cancellation in acylindrically hyperbolic groups
• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• Sur les Groupes Hyperboliques d’après Mikhael Gromov
• Geometry of the complex of curves I: Hyperbolicity
• Tree-graded spaces and asymptotic cones of groups
• Tight geodesics in the curve complex
• Property Γ and inner amenability
• Peripheral fillings of relatively hyperbolic groups
• Acylindrical accessibility for groups
• Dehn filling in relatively hyperbolic groups
• Metric Spaces of Non-Positive Curvature
• Orbit equivalence rigidity and bounded cohomology
• Deformation and rigidity for group actions and von Neumann algebras
• Hyperbolic Groups

Cited by

• On purely loxodromic actions
• Hierarchically hyperbolic spaces, I: Curve complexes for cubical groups
• Hyperbolic group [wikipedia]

Related

• Acylindrically hyperbolic groups
• Groups acting acylindrically on hyperbolic spaces
• Small cancellation in acylindrically hyperbolic groups
• Hyperfiniteness of boundary actions of acylindrically hyperbolic groups
• Largest acylindrical actions and stability in hierarchically hyperbolic groups