6.4 Examples and classification results

Hyperbolic groups and spaces are fundamental in geometric group theory, showcasing negative curvature in algebraic settings. They include free groups, surface groups, and small cancellation groups, each offering unique insights into group structure and geometry.

Classification of hyperbolic groups involves structural decomposition, boundary theory, and topological aspects. Key tools include the JSJ decomposition, Gromov boundary, and quasi-isometry invariance. These concepts help categorize and understand the diverse landscape of hyperbolic groups.

Hyperbolic Groups and Spaces

Fundamental Examples and Properties

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• Hyperbolic space Hn embodies constant negative curvature and unique geometric properties, serving as the archetypal hyperbolic metric space
• Free groups of rank ≥ 2 exemplify hyperbolic groups with exponential growth and strong geometric characteristics (tree-like structure)
• Surface groups of genus ≥ 2 represent fundamental groups of closed surfaces with negative Euler characteristic, demonstrating hyperbolicity in low-dimensional topology
• Small cancellation groups satisfying C'(1/6) or C'(1/4)-T(4) conditions form a significant class of hyperbolic groups, often used in constructing counterexamples
• Fundamental groups of compact hyperbolic manifolds bridge group theory and low-dimensional topology, providing concrete geometric realizations of hyperbolic groups

Gromov's Contributions and Boundary Theory

• Word-hyperbolic groups, introduced by Gromov, generalize negative curvature to finitely generated groups, revolutionizing geometric group theory
• The Gromov boundary of a hyperbolic group or space illuminates its asymptotic geometry and dynamics
  • For hyperbolic plane H^2, the boundary is a circle
  • For free groups, the boundary is a Cantor set
• The Rips complex encodes large-scale geometry of hyperbolic groups, serving as a key classification tool
  • Simplicial complex constructed from the group's Cayley graph
  • Captures quasi-isometry invariant properties of the group

Classification of Hyperbolic Groups

Structural Decomposition and Invariants

• JSJ decomposition theorem provides a structure theory for hyperbolic groups
  • Decomposes groups into simpler pieces along elementary subgroups
  • Analogous to JSJ decomposition of 3-manifolds
• Torsion in hyperbolic groups follows strict constraints
  • Finite subgroups have bounded order (Torsion Theorem)
  • Act with fixed points on the Gromov boundary
• Geometric dimension of a hyperbolic group correlates closely with its cohomological dimension
  • Provides important invariants for classification
  • Example Virtual cohomological dimension of a Fuchsian group equals 2
• Quasi-convex subgroups play a crucial role in understanding hyperbolic group structure
  • Inherit hyperbolicity from the ambient group
  • Examples include finite index subgroups and maximal cyclic subgroups

Topological and Geometric Aspects

• Cannon conjecture proposes a topological characterization of hyperbolic groups with sphere boundary
  • Conjectures they are fundamental groups of hyperbolic 3-manifolds
  • Remains one of the major open problems in geometric group theory
• Quasi-isometry invariance serves as a fundamental principle in hyperbolic group classification
  • Allows for flexibility in geometric representations
  • Example Quasi-isometric groups have homeomorphic Gromov boundaries
• Geometrization Theorem for 3-manifolds, proved by Perelman, classifies hyperbolic 3-manifolds and their fundamental groups
  • Resolves Thurston's Geometrization Conjecture
  • Provides a complete topological classification of 3-manifolds

Classification Theorems for Hyperbolic Groups

Structural Classification Results

• Stallings' theorem on groups with infinitely many ends classifies certain hyperbolic groups
  • Characterizes groups that split over finite subgroups
  • Example Free product of two infinite groups has infinitely many ends
• Sela's isomorphism problem for torsion-free hyperbolic groups provides a theoretical algorithm
  • Determines when two such groups are isomorphic
  • Utilizes JSJ decompositions and Makanin-Razborov diagrams
• Rips machine, developed by Rips and Sela, proves structure theorems about hyperbolic groups
  • Particularly effective for groups with small rank
  • Analyzes actions on real trees to derive structural information

Extensions and Combinations

• Theory of relatively hyperbolic groups extends classification results to broader classes
  • Includes fundamental groups of finite-volume hyperbolic manifolds
  • Example Fundamental group of a hyperbolic knot complement is hyperbolic relative to its cusp subgroup
• Bestvina and Feighn's combination theorem provides conditions for hyperbolicity
  • Applies to amalgamated free products and HNN extensions of hyperbolic groups
  • Example Amalgamated product of hyperbolic groups over malnormal quasiconvex subgroups is hyperbolic
• Classification of surface groups relies on Nielsen-Thurston classification
  • Utilizes theory of measured laminations
  • Classifies surface homeomorphisms as periodic, reducible, or pseudo-Anosov

Applications of Classification Results

Algorithmic and Structural Applications

• Word problem solution in hyperbolic groups based on Dehn's algorithm
  • Relies on classification of hyperbolic groups as having linear Dehn functions
  • Provides efficient algorithms for word problem and conjugacy problem
• Classification results applied to study automorphism groups of hyperbolic groups
  • Investigates rigidity phenomena and outer automorphism groups
  • Example Out(Fn) for free groups Fn exhibits interesting geometric and algebraic properties
• CAT(0) cube complex theory applies classification results to understand group actions
  • Developed by Sageev and others for non-positively curved spaces
  • Example Wise's work on special cube complexes and the resolution of the Virtual Haken Conjecture

Broader Implications in Group Theory

• Quasi-isometric rigidity results enable solution of quasi-isometry classification problems
  • Example Quasi-isometric rigidity of symmetric spaces of non-compact type
• Random group theory often utilizes hyperbolic group classification results
  • Particularly relevant in the Gromov density model
  • Example At density < 1/2, random groups are almost surely hyperbolic
• Classification theorems applied to boundary actions and amenability studies
  • Connects geometric group theory with ergodic theory and dynamics
  • Example Characterization of amenable hyperbolic groups as virtually cyclic groups
• Hyperbolic group structure results used to study profinite completions
  • Addresses problems in geometric and combinatorial group theory
  • Example Residual finiteness of hyperbolic groups and its implications for the profinite topology