3.2 Word metric and geodesics

Word metric and geodesics are key concepts in geometric group theory. They measure distance between group elements using generators and provide insights into group structure. These tools transform abstract groups into geometric spaces, enabling powerful analysis techniques.

Cayley graphs visually represent word metrics, with edges corresponding to generators. Geodesics in these graphs are shortest paths between vertices, revealing important group properties. This geometric perspective connects algebra to topology and computer science, opening new avenues for group study.

Word Metric on Groups

Definition and Basic Properties

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• Word metric on a group G with generating set S measures minimum number of generators needed to express one group element in terms of another
• For elements g, h in G, word metric d(g,h) equals length of shortest word in generators S representing g^(-1)h
• Word metric satisfies axioms of a metric space
  • Non-negativity
  • Symmetry
  • Triangle inequality
• Choice of generating set affects word metric (different generating sets can lead to different distances between same pair of elements)
• Word length of element g, denoted |g|, represents distance from g to identity element e using word metric
• Word metric demonstrates left-invariance (d(ag,ah) = d(g,h) for all a,g,h in G)

Visual Representation and Applications

• Cayley graph of group visually represents word metric (edge lengths correspond to generator lengths)
• Word metric plays crucial role in geometric group theory
  • Helps analyze group structure through geometric lens
  • Enables study of growth rates and large-scale structure of groups
• Applications extend to various mathematical fields
  • Topology (fundamental groups of manifolds)
  • Computer science (analysis of algorithms on groups)
  • Cryptography (group-based cryptosystems)

Computation and Challenges

• Computing word metric distance between elements g and h involves finding shortest word in generators representing g^(-1)h
• Efficient algorithms for word metric distance computation
  • Breadth-first search in Cayley graph
  • Dynamic programming techniques
• Computation complexity varies between finite and infinite groups
  • Finite groups may allow exhaustive search
  • Infinite groups often require more sophisticated methods
• Specialized algorithms exist for certain group classes
  • Free groups (use normal forms)
  • Abelian groups (use Smith normal form)
• Word problem (determining if given word represents identity) closely relates to word metric distance computation
• Some groups present computational challenges
  • Braid groups and mapping class groups often require approximation algorithms
  • Complexity of computation provides insights into group's algorithmic and geometric properties

Geodesics in Cayley Graphs

Fundamental Concepts

• Geodesic in Cayley graph represents path between two vertices realizing minimum distance in word metric
• Geodesics correspond to minimal length words in generators expressing one group element in terms of another
• Multiple geodesics may exist between two vertices (represent different minimal length words for same group element)
• Geodesic distance between vertices in Cayley graph equals word metric distance between corresponding group elements
• Geodesics in Cayley graphs not necessarily unique (multiple paths of minimal length may connect two vertices)

Importance in Geometric Group Theory

• Study of geodesics in Cayley graphs fundamental to understanding geometric properties of groups
  • Provides insights into growth rates
  • Reveals large-scale structure of groups
• Geodesics help classify groups based on geometric behavior
  • Hyperbolic groups (negatively curved)
  • CAT(0) groups (non-positively curved)
• Analysis of geodesics contributes to solving algorithmic problems in groups
  • Word problem
  • Conjugacy problem
  • Isomorphism problem

Special Properties in Hyperbolic Groups

• Geodesics in Cayley graphs of hyperbolic groups exhibit unique characteristics
  • Fellow-traveling property (geodesics with close endpoints remain close)
  • Quasi-geodesic stability (quasi-geodesics stay close to true geodesics)
• These properties lead to powerful algorithms for hyperbolic groups
  • Linear-time solutions to word problem
  • Efficient computation of canonical representatives
• Hyperbolic group geodesics provide geometric intuition for more general concepts
  • Relatively hyperbolic groups
  • Acylindrically hyperbolic groups

Distances in Word Metric

Computation Methods

• Word metric distance computation between elements g and h requires finding shortest word in generators representing g^(-1)h
• Efficient algorithms for distance computation
  • Breadth-first search in Cayley graph (works well for finitely generated groups)
  • Dynamic programming techniques (useful for groups with nice normal forms)
• Computation complexity varies between finite and infinite groups
  • Finite groups may allow exhaustive search
  • Infinite groups often require more sophisticated methods (heuristic search, approximation algorithms)

Group-Specific Algorithms

• Free groups utilize normal forms for efficient distance computation
  • Reduced words provide unique representatives
  • Distance equals length of reduced word for g^(-1)h
• Abelian groups employ Smith normal form for distance calculations
  • Transform generating set to standard basis
  • Distance becomes L1 norm in transformed coordinates
• Certain classes of groups have specialized distance algorithms
  • Coxeter groups (use reduced words and exchange condition)
  • Braid groups (use normal forms, but exact computation can be intractable)

Computational Challenges and Implications

• Word problem closely related to distance computation
  • Solving word problem equivalent to computing distance to identity element
  • Groups with undecidable word problem present fundamental obstacles to exact distance computation
• Some groups require approximation algorithms for practical distance computation
  • Mapping class groups (use subsurface projection techniques)
  • Out(Fn) (employ relative train track representatives)
• Complexity of distance computation provides insights into group structure
  • Polynomial-time algorithms suggest nice geometric or algebraic properties
  • Hardness results can indicate complex group structure or geometry

Properties of Word Metric

Symmetry and Invariance

• Word metric exhibits symmetry (d(g,h) = d(h,g) for all g,h in G)
  • Follows from group axioms and metric definition
  • Reflects reversibility of paths in Cayley graph
• Translation invariance characterizes word metric (d(ag,ah) = d(g,h) for all a,g,h in G)
  • Demonstrates homogeneity of group structure
  • Allows for consistent distance measurements throughout group

Geometric Structure and Invariants

• Word metric induces natural geometric structure on group
  • Transforms abstract group into concrete metric space
  • Enables application of geometric and topological techniques to group theory
• Large-scale geometry of group with word metric independent of finite generating set choice
  • Property known as quasi-isometry invariance
  • Allows for meaningful study of infinite groups up to geometric equivalence
• Growth function serves as fundamental geometric invariant
  • Measures number of elements within given word metric distance from identity
  • Classifies groups into different growth types (polynomial, exponential, intermediate)

Connections to Other Mathematical Areas

• Word metric defines important geometric concepts in groups
  • Quasi-geodesics (paths that approximate geodesics up to bounded error)
  • Hyperbolicity (negative curvature in large scale)
  • Asymptotic cones (limiting objects capturing large-scale geometry)
• Study of word metric properties connects group theory to various mathematical fields
  • Topology (fundamental groups of manifolds, geometric topology)
  • Differential geometry (nonpositively curved spaces, CAT(0) geometry)
  • Ergodic theory (random walks on groups, Poisson boundaries)