3.1 Construction and properties of Cayley graphs

Cayley graphs are powerful tools for visualizing group structure. They represent groups using vertices for elements and edges for generator actions, providing a geometric perspective on algebraic properties.

These graphs encode crucial information about groups, including connectivity, regularity, and distance metrics. They're essential in geometric group theory, offering insights into growth rates, word problems, and large-scale group properties.

Cayley graphs for groups

Fundamental concepts and construction

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• Cayley graph graphically represents group G with generating set S
  • Vertices correspond to group elements
  • Directed edges represent multiplication by generators
• Vertex set comprises all elements in group G
• Edge from vertex g to h exists if and only if h = gs for generator s in S
• Identity element serves as distinguished vertex
• Left-invariant structure (graph looks identical from any vertex)
• Locally finite for finite generating set S (each vertex has finite degree)
• Generating set S determines graph structure
  • Different generating sets can produce distinct graphs for same group

Properties and characteristics

• Always connected (any element reachable from identity via generator multiplications)
• Regular graph (degree of each vertex equals number of generators in S)
• Undirected if generating set S is symmetric (closed under inverses)
• Distance between vertices corresponds to word length between group elements
• Diameter relates to maximum word length in group for generating set
• Finite for finite groups, infinite but locally finite for infinite groups
• Girth (shortest cycle length) linked to shortest non-trivial generator relation

Examples and applications

• Cayley graph of cyclic group $Z_6$ with generator $\{1\}$ forms a directed 6-cycle
• Cayley graph of dihedral group $D_4$ with generators $\{r,s\}$ (rotation, reflection) creates square with diagonal edges
• Used to visualize group actions (rotations, reflections of regular polygons)
• Applied in computer science for network design and parallel computing
• Employed in cryptography for analyzing group-based encryption schemes

Connectivity and regularity

Graph theoretic properties

• Connected nature ensures path exists between any two vertices
• Regular structure (constant vertex degree) simplifies analysis
• Vertex degree equals size of generating set |S|
• Symmetric generating sets yield undirected graphs
• Distance metric on vertices induces word metric on group elements
• Diameter represents worst-case distance between elements
• Girth relates to shortest non-trivial relation among generators

Geometric interpretations

• Cayley graphs embed groups into metric spaces
• Ball growth in graph corresponds to group growth function
• Hyperbolic groups have Cayley graphs with negative curvature properties
• Amenable groups have Cayley graphs with certain averaging properties
• Quasi-isometry between Cayley graphs preserves large-scale geometry
• Asymptotic cones capture limiting behavior of scaled Cayley graphs

Examples and applications

• Cayley graph of free group on two generators forms infinite 4-regular tree
• Cayley graph of $Z^2$ with standard generators yields infinite square grid
• Used to study random walks and diffusion processes on groups
• Applied in geometric group theory to classify groups by large-scale properties
• Employed in harmonic analysis to study Fourier transforms on groups

Encoding group structure

Automorphisms and symmetries

• Automorphism group contains subgroup isomorphic to original group G
• Left multiplication by group elements induces graph automorphisms
• Demonstrates left-invariance of Cayley graphs
• Cycles correspond to relations among generators in group presentation
• Growth rate of graph balls reflects group growth function
• Isomorphic Cayley graphs for different generating sets show quasi-isometry
• Cosets and subgroups visualized as subgraphs or quotients

Algebraic properties

• Word problem solvability related to decidability of graph reachability
• Conjugacy classes represented by certain vertex orbits under automorphisms
• Normal subgroups correspond to equitable partitions of the graph
• Centralizers and normalizers reflected in vertex stabilizers
• Group extensions visualized through graph fibrations or coverings
• Commutator subgroup related to certain closed walks in the graph

Applications and examples

• Cayley graph of symmetric group $S_n$ encodes permutation structure
• Cayley graphs of Coxeter groups reflect geometric symmetries (reflections, rotations)
• Used to study growth rates and isoperimetric inequalities in groups
• Applied in geometric group theory to prove Gromov's theorem on groups of polynomial growth
• Employed in computational group theory for efficient algorithms on finitely generated groups

Generating sets vs Cayley graphs

Relationship and dependencies

• Different generating sets for same group can yield non-isomorphic Cayley graphs
• Resulting graphs guaranteed to be quasi-isometric
• Minimal generating sets often produce more efficient group structure representations
• Adding redundant generators increases vertex degree without altering group structure
• Generating set choice impacts graph properties (diameter, girth, growth rate)
• Normal subgroup Cayley graphs relate to quotient graphs of larger group
• Cayley color graphs encode detailed generating set structure through edge coloring

Comparative analysis

• Sparse generating sets tend to produce graphs with larger diameter
• Redundant generating sets can decrease graph diameter but increase local complexity
• Symmetric generating sets yield undirected graphs, simplifying certain analyses
• Generating sets with relations produce graphs with specific cycle structures
• Choice of generators affects word length metric and geodesics in graph
• Trade-off between simplicity of generators and efficiency of resulting graph

Examples and applications

• Cayley graph of $Z$ with generator $\{1\}$ forms infinite line, while $\{1,2\}$ creates infinite comb graph
• Cayley graph of free group changes drastically with addition of relations
• Used to study random generation and probabilistic group properties
• Applied in geometric group theory to analyze asymptotic cones and growth series
• Employed in combinatorial group theory to solve word and conjugacy problems efficiently