1.2 Fundamental concepts and definitions

Geometric Group Theory bridges algebra and geometry, exploring groups through their actions on spaces. This intro focuses on fundamental concepts like group definitions, generators, and relations, laying the groundwork for understanding group structure and behavior.

Cayley graphs visually represent groups, offering insights into their properties. We'll also explore metrics, geodesics, and quasi-isometries, which help compare groups and analyze their large-scale geometry. These tools are essential for studying more advanced topics in the field.

Groups, generators, and relations

Fundamental group concepts

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• Group consists of a set of elements and a binary operation satisfying four axioms
  • Closure ensures the operation always produces another group element
  • Associativity allows regrouping of elements without changing the result
  • Identity element leaves other elements unchanged when combined
  • Inverse element exists for each element, producing the identity when combined
• Order of a group refers to the number of elements (finite or infinite)
• Free group on a set of generators has no relations beyond group axioms
  • Allows unrestricted combination of generators
  • Serves as a building block for more complex groups

Generators and relations

• Generators produce all group elements through repeated application of the group operation
  • Example: integers under addition generated by 1 and -1
  • Example: rotational symmetries of a square generated by 90-degree rotation
• Relations equations between words in the generators hold true for all group elements
  • Example: $r^4 = e$ for 90-degree rotation $r$ of a square
  • Example: $aba^{-1}b^{-1} = e$ for commuting elements $a$ and $b$
• Group presentation describes a group using generators and relations
  • Written as ⟨S|R⟩, where S generators and R relations
  • Example: ⟨a,b | a^2=b^2=(ab)^2=e⟩ describes the Klein four-group
• Tietze transformations modify group presentations while preserving the group structure
  • Adding or removing redundant generators
  • Adding or removing consequences of existing relations
  • Example: ⟨a,b | a^3=b^2=e⟩ to ⟨a,b,c | a^3=b^2=c^2=abc=e⟩

Cayley graphs and groups

Cayley graph construction and properties

• Cayley graph visually represents group structure
  • Vertices correspond to group elements
  • Edges represent multiplication by generators
  • Example: Cayley graph of $Z_4$ with generator 1 forms a square
• Cayley graph depends on chosen generating set
  • Different generators can produce distinct graphs for the same group
  • Example: $Z$ with generator 1 forms an infinite line, while generators 2 and 3 form a more complex graph
• Cayley graph always connected, regular, and vertex-transitive
  • Connectedness reflects generators producing all elements
  • Regularity shows each element acted on by same number of generators
  • Vertex-transitivity indicates group's homogeneity
• Loops and multiple edges in Cayley graph correspond to specific generator properties
  • Loops indicate generators of order 1 (identity element)
  • Multiple edges show generators of order 2 (self-inverse elements)

Cayley graph analysis and applications

• Distance between vertices represents word length transforming one element to another
  • Provides geometric interpretation of algebraic properties
  • Example: in $Z$, distance between vertices 3 and 7 equals 4, corresponding to multiplication by generator 1 four times
• Automorphisms of group correspond to Cayley graph symmetries
  • Reveals structural information about the group
  • Example: rotational symmetry of Cayley graph for cyclic group reflects group's cyclic nature
• Growth rate of Cayley graph measures vertex increase with distance from identity
  • Important group invariant independent of generating set
  • Example: polynomial growth for abelian groups, exponential for free groups

Word metric, geodesics, and quasi-isometries

Word metric and geodesics

• Word metric defines distance function on group elements
  • Measures minimum number of generators needed to express one element in terms of another
  • Example: in free group on two generators a and b, distance between aba and ba^2 equals 3 (multiply by a^-1, then b^-1, then a)
• Geodesic path of minimal length between two vertices in Cayley graph
  • Corresponds to shortest word representation in group
  • Example: in $Z^2$, geodesic from (0,0) to (3,4) straight line, representing word a^3b^4
• Dehn function quantifies difficulty of word problem
  • Measures area needed to fill loops in Cayley graph
  • Example: linear Dehn function for hyperbolic groups, quadratic for $Z^2$

Quasi-isometries and invariants

• Quasi-isometries preserve large-scale geometry while allowing bounded small-scale distortion
  • Map between metric spaces satisfying specific distance-preserving conditions
  • Example: embedding of $Z$ into $R$ quasi-isometry, despite different local structures
• Groups quasi-isometric if Cayley graphs (with finite generating sets) quasi-isometric as metric spaces
  • Captures notion of groups having similar large-scale geometry
  • Example: $Z$ and $R$ quasi-isometric, reflecting similar linear structure
• Quasi-isometry invariants properties preserved under quasi-isometries
  • Include growth rate, number of ends, hyperbolicity
  • Example: exponential growth rate preserved between quasi-isometric groups
• Švarc-Milnor lemma connects group actions and quasi-isometries
  • Group acting properly and cocompactly on metric space quasi-isometric to that space
  • Example: fundamental group of compact manifold quasi-isometric to universal cover

Hyperbolic vs CAT(0) vs Automatic groups

Hyperbolic and CAT(0) groups

• Hyperbolic group finitely generated group with Cayley graph satisfying hyperbolic space-like conditions
  • Generalizes negatively curved spaces to group theory context
  • Example: free groups hyperbolic, reflecting tree-like structure of Cayley graph
• δ-hyperbolic condition requires "thin" geodesic triangles
  • Any point on one side of triangle close to one of other two sides
  • Hyperbolicity constant δ quantifies this thinness
  • Example: trees 0-hyperbolic, as geodesic triangles degenerate to tripods
• CAT(0) spaces generalize non-positive curvature
  • Geodesic triangles "no fatter" than Euclidean counterparts
  • Example: Euclidean plane CAT(0) but not hyperbolic
• CAT(0) group acts properly and cocompactly by isometries on CAT(0) space
  • Captures geometric properties of non-positively curved spaces
  • Example: fundamental groups of compact non-positively curved manifolds CAT(0)

Automatic groups and recognition

• Automatic groups admit finite state automaton recognizing normal form for elements
  • Provides efficient solution to word problem
  • Example: finite groups automatic, with automaton simply listing all elements
• Fellow traveler property key characteristic of automatic groups
  • Nearby points in Cayley graph have nearby geodesic paths to identity
  • Example: in $Z^2$, paths to (n,m) and (n+1,m) stay within distance 1 of each other
• Automatic groups include hyperbolic groups and many important examples
  • Braid groups automatic, despite complex geometric structure
  • Mapping class groups of surfaces automatic, reflecting combinatorial nature
• Automaticity provides computational tools for studying groups
  • Efficient algorithms for solving word and conjugacy problems
  • Example: automatic structure for braid groups yields polynomial-time solution to word problem