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