4.3 Geometric properties of groups

Geometric properties of groups offer a powerful lens to understand their structure and behavior. By viewing groups as metric spaces through Cayley graphs, we can analyze their growth, hyperbolicity, and other geometric features. These properties reveal deep insights into group structure and behavior.

Quasi-isometries provide a framework for comparing groups geometrically, preserving large-scale properties while allowing local distortions. This approach yields invariants like growth rate and number of ends, connecting geometric intuition to fundamental group characteristics. Understanding these connections is key to grasping the chapter's core concepts.

Group Structure and Geometry

Word Metric and Geometric Interpretation

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• Word Metric defines natural geometric structure on groups
  • Distance between elements measured by length of shortest word representing their product
  • Provides way to visualize group elements in geometric space
• Finitely generated groups viewed as metric spaces through Cayley graphs
  • Cayley graphs offer geometric interpretation of group operations
  • Vertices represent group elements, edges represent generator actions
• Growth function measures expansion of balls in Cayley graph
  • Reflects both algebraic and geometric properties of group
  • Examples: polynomial growth (nilpotent groups), exponential growth (free groups)
• Isoperimetric inequalities connect subset size to boundary size
  • Bridge between algebraic structure and geometric notions of expansion
  • Crucial in studying Dehn functions and word problem solvability

Geometric Properties and Group Structure

• Hyperbolicity profoundly impacts algebraic structure
  • Implies word problem solvability
  • Characterizes groups with negatively curved Cayley graphs (hyperbolic groups)
• CAT(0) property yields important structural insights
  • Guarantees certain finiteness properties
  • Examples: Coxeter groups, right-angled Artin groups
• Group actions on metric spaces reveal deep connections
  • Proper and cocompact actions especially significant
  • Švarc-Milnor lemma links such actions to quasi-isometries

Quasi-isometries and Large-scale Geometry

• Quasi-isometries preserve large-scale geometric properties
  • Allow for local distortions
  • Provide framework for comparing groups geometrically
• Quasi-isometry invariants capture fundamental group properties
  • Growth rate (polynomial vs exponential)
  • Number of ends
  • Hyperbolicity
  • Asymptotic dimension

Cayley Graphs of Groups

Structure and Representation

• Cayley graphs visually represent group structure
  • Vertices correspond to group elements
  • Edges represent actions of generators
• Generating set choice affects graph geometry
  • Large-scale properties remain invariant under different finite generating sets
  • Example: $\mathbb{Z}$ with generators $\{1\}$ vs $\{2,3\}$ yield different graphs but same large-scale line structure
• Geodesics in Cayley graphs correspond to minimal length words
  • Connect algebraic and geometric notions of distance
  • Example: In free group on two generators, geodesic path represents reduced word

Graph Properties and Group Characteristics

• Vertex valence determined by generator count and inverses
  • Reflects complexity of group presentation
  • Example: Free group on n generators has vertex valence 2n
• Cayley graphs of abelian groups exhibit grid-like structures
  • Example: $\mathbb{Z}^2$ with standard generators forms square grid
• Free group Cayley graphs resemble trees
  • Illustrates lack of relations between generators
  • Example: Free group on two generators has 4-valent tree as Cayley graph
• Subgroups correspond to subgraphs of Cayley graph
  • Geometry provides insights into normality and finite index
  • Example: Index-2 subgroup appears as bipartite structure in Cayley graph

Advanced Concepts

• Asymptotic cone captures large-scale geometry
  • Allows study of asymptotic properties
  • Example: Asymptotic cone of $\mathbb{Z}^2$ is the Euclidean plane
• Cayley graphs enable visualization of group theoretic concepts
  • Cosets, normalizers, centralizers can be identified in graph structure
  • Example: Left cosets of a subgroup form disjoint "copies" of the subgroup's Cayley graph

Quasi-isometry Type of Groups

Quasi-isometry Invariants

• Growth rate preserved under quasi-isometries
  • Distinguishes fundamentally different group types
  • Examples: Polynomial growth (nilpotent groups) vs exponential growth (free groups)
• Ends of groups measure "infinities" in Cayley graph
  • Preserved under quasi-isometries
  • Provide coarse classification (0, 1, 2, or infinitely many ends)
• Hyperbolicity characterizes negatively curved spaces at large scales
  • Quasi-isometry invariant property
  • Examples: Free groups, fundamental groups of hyperbolic manifolds
• Asymptotic dimension generalizes topological dimension
  • Invariant under quasi-isometries
  • Example: $\mathbb{Z}^n$ has asymptotic dimension n

Quasi-isometric Rigidity

• Gromov's theorem for lattices in semisimple Lie groups
  • Shows quasi-isometry classes can uniquely determine groups up to finite index
  • Example: Quasi-isometry between cocompact lattices in same Lie group implies commensurability
• Quasi-isometric rigidity of symmetric spaces
  • Quasi-isometries between symmetric spaces of non-compact type are close to isometries
  • Implies strong restrictions on quasi-isometries between their lattices

Asymptotic Methods

• Asymptotic cones capture "view from infinity" of group geometry
  • Tool for studying quasi-isometry invariants
  • Example: Asymptotic cone of hyperbolic group is an $\mathbb{R}$-tree
• Limit groups and asymptotic cones
  • Connect geometric group theory to model theory
  • Provide insights into equations over groups

Geometric Techniques for Group Theory

Fundamental Lemmas and Inequalities

• Švarc-Milnor lemma connects group actions to quasi-isometries
  • Allows geometric properties of spaces to inform group structure
  • Example: Fundamental group of compact manifold quasi-isometric to its universal cover
• Dehn functions measure word problem difficulty
  • Interpreted through isoperimetric inequalities in Cayley graphs
  • Examples: Linear Dehn function (hyperbolic groups), quadratic (CAT(0) groups)

Applications to Classical Problems

• Novikov's conjecture resolved for hyperbolic groups
  • Uses geometric methods to prove topological invariance of higher signatures
• Thurston's geometrization conjecture
  • Geometric group theory techniques crucial in Perelman's proof
  • Classifies 3-manifolds using geometric structures

Boundaries and Dynamics

• Gromov boundary of hyperbolic groups
  • Powerful tool for understanding group dynamics
  • Example: Boundary of free group is Cantor set
• Rigidity phenomena in geometric group theory
  • Mostow rigidity theorem uses boundary behavior
  • Example: Isomorphism between fundamental groups of closed hyperbolic manifolds implies isometry

Specialized Geometric Approaches

• CAT(0) geometry provides insights into group structure
  • Applies to Coxeter groups, right-angled Artin groups
  • Example: CAT(0) cube complexes used to solve virtual Haken conjecture
• Hyperbolic geometry crucial in random group theory
  • Studies generic properties of finitely presented groups
  • Example: Random groups at density < 1/2 are hyperbolic with overwhelming probability