11.2 Fundamental groups of 3-manifolds

Fundamental groups of 3-manifolds are key to understanding their structure. They capture essential information about loops and help classify these spaces. By studying these groups, we can uncover deep connections between topology, geometry, and algebra in 3D.

Techniques like Van Kampen's theorem and covering spaces let us compute and analyze these groups. The Geometrization Conjecture, now proven, shows how fundamental groups relate to geometric structures, revolutionizing our understanding of 3-manifolds and their classification.

Fundamental Groups of 3-Manifolds

Topological Invariants and Computational Techniques

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• Fundamental group captures essential structure of loops in topological spaces, particularly relevant for 3-manifolds
• Van Kampen's theorem computes fundamental groups of 3-manifolds by decomposing them into simpler pieces
• Seifert-van Kampen theorem computes fundamental groups of spaces obtained by gluing simpler spaces along common subspaces
• Dehn's lemma and loop theorem analyze fundamental group of a 3-manifold in terms of its boundary
• Covering spaces reveal relationship between fundamental group and topology of a 3-manifold
• Combinatorial group theory techniques used in computing fundamental groups
  • Presentations of groups
  • Free products with amalgamation
• Geometric group theory techniques provide insights into fundamental group structure
  • Study of group actions on spaces
  • Quasi-isometries

Advanced Concepts and Applications

• Poincaré conjecture (now theorem) states any closed, simply connected 3-manifold homeomorphic to 3-sphere
• Prime decomposition theorem classifies 3-manifolds in terms of prime factors, reflected in fundamental group structure
• Thurston's geometrization program provides classification framework based on fundamental group properties
• Hyperbolic 3-manifolds characterized by fundamental groups acting discretely by isometries on hyperbolic 3-space
• Seifert fibered spaces exhibit specific algebraic properties related to circle actions in fundamental groups
• JSJ decomposition theorem decomposes 3-manifolds along incompressible tori, reflected in fundamental group structure
• Virtual properties of 3-manifold groups crucial in classification
  • Virtually fibered
  • Virtual positive first Betti number

Classifying 3-Manifolds by Fundamental Groups

Geometric Structures and Classification

• Thurston's eight geometries provide framework for understanding 3-manifold geometric structure
  • $S^3, E^3, H^3, S^2 \times R, H^2 \times R, Nil, Sol, \widetilde{SL}(2,R)$
• Hyperbolic geometry plays central role
  • Many 3-manifolds admit hyperbolic structures
  • Characterized by discrete, torsion-free subgroups of $PSL(2,C)$
• Curvature in 3-manifolds related to fundamental group properties
  • Negative curvature often corresponds to hyperbolic behavior
• Seifert fibered spaces have fundamental groups with specific algebraic properties
  • Related to geometric structure
  • Often involve circle actions

Volume and Large-Scale Properties

• Relationship between hyperbolic 3-manifold volume and fundamental group properties
  • Growth rate of geodesics provides insights into geometry-group theory connection
• Study of ends of 3-manifolds and fundamental groups reveals large-scale geometry and topology
• Geometrization Conjecture (now proven) establishes deep connection between topology, geometry, and fundamental group
• Resolution of Geometrization Conjecture implies 3-manifold groups residually finite
• Classification of 3-manifolds with finite fundamental groups corresponds to spherical 3-manifolds
• Hyperbolic 3-manifolds have word-hyperbolic fundamental groups in sense of Gromov

Geometry and Fundamental Groups of 3-Manifolds

Geometric Properties and Group Structure

• Virtually Fibered Conjecture (now theorem) states every hyperbolic 3-manifold has finite-sheeted cover fibering over circle
  • Provides deep insight into fundamental group structure
• Geometrization Conjecture implies 3-manifold groups coherent
  • Every finitely generated subgroup finitely presented
• Application of geometrization resolves decision problems in 3-manifold groups
  • Word problem
  • Conjugacy problem
• Fundamental groups of hyperbolic 3-manifolds exhibit specific properties
  • Gromov hyperbolic
  • Residually finite
  • Virtually fibered

Topological and Algebraic Connections

• JSJ decomposition reflects in fundamental group structure
  • Decomposition along incompressible tori
  • Algebraic splitting of fundamental group
• Seifert fibered spaces have fundamental groups with specific algebraic properties
  • Central extensions of surface groups
  • Reflect circle actions on the manifold
• Prime decomposition of 3-manifolds corresponds to free product decomposition of fundamental groups
• Covering space theory connects fundamental groups to topology
  • Universal cover and deck transformations
  • Relation between subgroups and covering spaces

Geometrization Conjecture and Fundamental Groups

Implications for Group Properties

• Geometrization Conjecture proves every 3-manifold group residually finite
  • Important algebraic property for infinite groups
• 3-manifold groups proven coherent due to Geometrization
  • Finitely generated subgroups always finitely presented
• Word problem and conjugacy problem solvable for 3-manifold groups
  • Consequence of geometric decomposition
• Virtual properties of 3-manifold groups established
  • Virtually Haken
  • Virtually fibered (for hyperbolic case)

Geometric Decomposition and Group Structure

• Every closed, oriented 3-manifold decomposable into geometric pieces
  • Each piece modeled on one of Thurston's eight geometries
• Hyperbolic pieces correspond to word-hyperbolic fundamental groups
• Seifert fibered pieces have fundamental groups with specific algebraic structure
  • Often involve circle actions or central extensions
• JSJ decomposition reflected in fundamental group as graph of groups
  • Incompressible tori correspond to amalgamated products or HNN extensions
• Prime decomposition theorem corresponds to free product decomposition of fundamental group
  • Each prime factor contributes a factor to the free product