6.1 Definition and characterizations of hyperbolic spaces

Hyperbolic spaces are a key concept in geometric group theory, showcasing non-Euclidean geometry with constant negative curvature. They're characterized by diverging geodesics and unique properties like ideal points at infinity, setting them apart from Euclidean spaces.

These spaces can be defined and characterized in multiple ways, including thin triangles, Gromov's δ-hyperbolicity, and CAT(κ) conditions. Understanding these equivalences and their geometric properties is crucial for grasping the broader landscape of hyperbolic groups and spaces.

Hyperbolic spaces and their properties

Fundamental concepts and models

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• Hyperbolic spaces exemplify non-Euclidean geometry with constant negative curvature characterized by diverging geodesics
• Two-dimensional hyperbolic geometry represented through Poincaré disk model and upper half-plane model
• Unique properties include existence of ideal points at infinity and absence of parallel lines (Euclidean sense)
• Hyperbolic distance function defines metric structure differing from Euclidean distance
• Isometry group encompasses translations, rotations, and hyperbolic reflections preserving hyperbolic distance between points
• Gauss-Bonnet theorem for hyperbolic triangles relates area directly to angle defect

Advanced geometric properties

• Exponential growth of ball volumes contrasts with polynomial growth in Euclidean spaces, leading to distinct large-scale geometric behaviors
• Margulis Lemma relates injectivity radius to systole of closed geodesics
• Horoballs and horospheres provide insight into asymptotic behavior of geodesics and ideal boundary structure
• Isometries classified into elliptic, parabolic, and hyperbolic types, each with distinct fixed point properties
• Convexity properties of distance functions lead to results like Švarc-Milnor lemma, connecting group actions to quasi-isometries
• Geodesic flows exhibit strong ergodic properties (mixing and Patterson-Sullivan measure on boundary)
• Rigidity phenomena (Mostow rigidity for finite-volume hyperbolic manifolds) demonstrate deep connections between geometry and topology

Characterizations of hyperbolic spaces

Metric and geometric conditions

• Thin triangle condition defines hyperbolic spaces through δ-thin geodesic triangles for fixed δ > 0
• Gromov's δ-hyperbolicity condition uses four-point configurations for metric characterization
• CAT(κ) condition for κ < 0 offers comparison geometry approach relating to spaces of constant negative curvature
• Visibility axiom characterizes hyperbolic spaces through behavior of geodesic rays and existence of visibility points at infinity
• Exponential divergence functions quantify geodesic divergence, providing alternative characterization
• Ideal boundary offers topological characterization through compactification
• Quasi-geodesics exhibit stability properties, serving as alternative hyperbolicity characterization

Advanced characterizations and applications

• Margulis Lemma characterizes thin parts of hyperbolic manifolds through short closed geodesics
• Gromov product provides a way to measure how close two geodesic rays are at infinity
• Bounded cocompactness condition characterizes hyperbolic spaces through group actions
• Cannon's conjecture relates word-hyperbolic groups to 3-dimensional hyperbolic geometry
• Morse lemma for quasi-geodesics demonstrates stability of quasi-geodesics in hyperbolic spaces
• Asymptotic dimension provides a coarse geometric characterization of hyperbolic spaces
• Conformal dimension of the boundary at infinity offers a quasiconformal characterization

Equivalence of hyperbolic space definitions

Proving equivalence between core definitions

• Thin triangle condition and Gromov's δ-hyperbolicity equivalence established through geometric arguments and inequalities
• CAT(κ) condition for κ < 0 and thin triangle condition equivalence involves comparing geodesic triangles to hyperbolic plane counterparts
• Visibility axiom shown equivalent to thin triangle condition by constructing geodesic configurations and analyzing limiting behavior
• Exponential divergence of geodesics and δ-hyperbolicity equivalence requires analysis of distance growth rates between geodesic rays
• Quasi-geodesic stability proven equivalent to δ-hyperbolicity through fellow-traveling geodesics construction and geometric property analysis
• Ideal boundary characterization equivalence involves geometry and topology techniques (horofunctions and Busemann functions)

Advanced equivalence proofs and connections

• Rips complex construction demonstrates equivalence between δ-hyperbolicity and coarse negative curvature
• Floyd boundary equivalence to Gromov boundary for word-hyperbolic groups involves analyzing word metrics
• Equivalence between CAT(-1) spaces and visibility manifolds established through geometric and topological arguments
• Gromov's characterization of hyperbolic groups in terms of linear isoperimetric inequality proven equivalent to δ-hyperbolicity
• Equivalence between rapid decay property and hyperbolicity for finitely generated groups involves analysis of group cohomology
• Cannon's conjecture relates word-hyperbolicity to 3-dimensional hyperbolic geometry through quasiconformal analysis of group boundaries

Geometry and metrics of hyperbolic spaces

Metric properties and structures

• Hyperbolic distance function satisfies specific axioms differing from Euclidean distance
• Triangle inequality in hyperbolic spaces exhibits stronger form known as Ptolemaic inequality
• Busemann functions provide a way to study asymptotic behavior of geodesic rays
• Horofunctions generalize notion of distance to ideal points at infinity
• Gromov product measures how long two geodesics stay close together
• Visual metrics on the boundary at infinity encode geometric information about the space
• Quasihyperbolic metric provides a way to study non-complete hyperbolic spaces

Geometric structures and dynamics

• Geodesic flows in hyperbolic spaces exhibit mixing and ergodicity properties
• Horospheres and horoballs play crucial role in studying cusps of hyperbolic manifolds
• Convex subsets of hyperbolic spaces have unique properties (totally geodesic submanifolds)
• Isometry classification (elliptic, parabolic, hyperbolic) relates to fixed point behavior
• Limit set of a discrete group acting on hyperbolic space encodes dynamical information
• Patterson-Sullivan measures on the boundary provide invariant measures for the geodesic flow
• Geometric group actions on hyperbolic spaces lead to rich theory of hyperbolic groups