6.2 Properties of hyperbolic groups

Hyperbolic groups are finitely generated groups with Cayley graphs resembling hyperbolic space geometry. They're defined by the δ-hyperbolic condition, which requires δ-thin geodesic triangles in the Cayley graph for a fixed δ ≥ 0.

These groups have fascinating properties, including linear isoperimetric inequality, exponential growth rate, and solvable word and conjugacy problems. They also possess a boundary at infinity and follow the Tits alternative, making them a rich subject in geometric group theory.

Hyperbolic Groups

Definition and Core Properties

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• Hyperbolic groups consist of finitely generated groups whose Cayley graphs mimic hyperbolic space geometry
• δ-hyperbolic condition defines hyperbolic groups through δ-thin geodesic triangles in the Cayley graph for a fixed δ ≥ 0
• Word-hyperbolic groups satisfy linear isoperimetric inequality bounding van Kampen diagram areas linearly by boundary length
• Exponential growth rate characterizes hyperbolic groups with element count in Cayley graph balls growing exponentially with radius
• Solvable word and conjugacy problems in hyperbolic groups use linear and quadratic time algorithms respectively
• Boundary at infinity encodes asymptotic group information as a compact, metrizable space
• Tits alternative classifies hyperbolic groups as either virtually cyclic or containing non-abelian free subgroups

Geometric and Algorithmic Aspects

• Cayley graph of hyperbolic groups forms a δ-hyperbolic metric space when equipped with word metric
• Linear isoperimetric function characterizes word problem efficiency and van Kampen diagram geometry
• Finite asymptotic dimension generalizes covering dimension to large-scale geometry impacting Novikov conjecture
• Rational function growth series encodes growth rate and algebraic structure information
• Finiteness properties include finite presentation and type F∞, relating to cohomological dimensions
• Finite outer automorphism group for most hyperbolic groups reflects hyperbolic geometry rigidity
• Quasi-convex subgroups inherit hyperbolicity and finite generation, aiding subgroup structure analysis

Hyperbolic Group Theorems

Fundamental Theorems

• Švarc-Milnor Lemma establishes quasi-isometry between groups acting properly and cocompactly on proper geodesic metric spaces and the spaces themselves
• Gromov's theorem equates hyperbolic groups to those acting properly and cocompactly by isometries on proper δ-hyperbolic spaces
• Morse Lemma demonstrates bounded distance between quasi-geodesics and true geodesics in δ-hyperbolic spaces
• Stability of Quasi-geodesics extends Morse Lemma showing uniform closeness of quasi-geodesics with common endpoints
• Bounded Coset Penetration proves continued fellow-traveling of long-distance quasi-geodesics in hyperbolic groups
• Combination Theorem provides hyperbolicity conditions for amalgamated free products and HNN extensions of hyperbolic groups
• Rips Machine algorithm solves word problem in hyperbolic groups in linear time

Advanced Theorems and Applications

• Quasi-isometries between hyperbolic spaces induce boundary at infinity homeomorphisms, connecting group actions and boundary dynamics
• Cannon's conjecture links hyperbolic groups with 2-sphere boundaries to virtual surface groups, bridging group theory and 3-manifold topology
• Quasi-isometry classification of hyperbolic groups equates to topological classification of boundaries up to quasi-Möbius homeomorphism
• Hyperbolic groups naturally arise as fundamental groups of negatively curved closed manifolds, connecting group theory and differential geometry
• Malnormality of quasi-convex subgroups relates to boundary at infinity action dynamics
• Novikov conjecture implications stem from finite asymptotic dimension property of hyperbolic groups
• Rational growth series provides insights into algebraic structure and growth patterns of hyperbolic groups

Hyperbolic Groups vs Spaces

Connections and Equivalences

• Cayley graphs of hyperbolic groups form δ-hyperbolic metric spaces with word metric
• Group actions on hyperbolic spaces induce boundary homeomorphisms through quasi-isometries
• Hyperbolic group actions on Cayley graphs extend to continuous, minimal, and expansive actions on the boundary at infinity
• Visual boundary of CAT(0) spaces with geometric hyperbolic group actions homeomorphic to group's Gromov boundary
• Fundamental groups of negatively curved closed manifolds exemplify natural occurrences of hyperbolic groups
• Quasi-isometry classification of hyperbolic groups equates to topological classification of boundaries
• δ-hyperbolic condition in spaces translates to thin triangles property in group Cayley graphs

Applications and Implications

• Boundary dynamics analysis provides insights into algebraic properties (malnormality of quasi-convex subgroups)
• Cannon's conjecture bridges hyperbolic group theory with 3-manifold topology
• CAT(0) space theory intersects with hyperbolic group theory through boundary considerations
• Negatively curved manifold study benefits from hyperbolic group theory applications
• Large-scale geometry concepts (asymptotic dimension) apply to both hyperbolic groups and spaces
• Quasi-isometry invariants of spaces translate to group invariants for hyperbolic groups
• Boundary at infinity serves as a unifying concept between hyperbolic spaces and groups, encoding asymptotic information

Properties of Hyperbolic Groups

Algebraic Properties

• Finite generation defines hyperbolic groups, ensuring a finite generating set exists
• Linear isoperimetric inequality characterizes word-hyperbolic groups, bounding van Kampen diagram areas
• Exponential growth rate distinguishes hyperbolic groups, with element count in Cayley graph balls growing exponentially
• Solvable word and conjugacy problems utilize efficient algorithms (linear and quadratic time respectively)
• Tits alternative classifies hyperbolic groups as either virtually cyclic or containing free subgroups
• Finite presentation and type F∞ finiteness properties relate to cohomological dimensions
• Finite outer automorphism group for non-elementary hyperbolic groups reflects geometric rigidity

Geometric and Topological Properties

• δ-thin triangles in Cayley graphs define the fundamental δ-hyperbolic condition
• Boundary at infinity forms a compact, metrizable space encoding asymptotic group information
• Quasi-convex subgroups inherit hyperbolicity and finite generation from parent group
• Finite asymptotic dimension generalizes covering dimension to large-scale group geometry
• Rational function growth series encodes both algebraic structure and growth rate information
• Cayley graph equipped with word metric forms a proper δ-hyperbolic space
• Group action on boundary at infinity is continuous, minimal, and expansive