⭕Geometric Group Theory Unit 2 – Free Groups and Presentations

What Are Free Groups?

• Algebraic structures consisting of a set of elements and a binary operation (group operation) that satisfies the group axioms (closure, associativity, identity, and inverses)
• Generated by a set of elements (generators) with no relations among them other than the group axioms
  • Generators can be combined using the group operation to form any element in the free group
• Characterized by the property of being freely generated, meaning there are no additional constraints or relations imposed on the generators
• Play a fundamental role in group theory as they serve as building blocks for constructing more complex groups
• Can be thought of as the most general type of group generated by a given set of elements
• Useful in studying properties and structures that are common to all groups
• Provide a framework for understanding the concept of group presentations and their relationships to other areas of mathematics (topology, geometry)

Building Blocks: Generators and Relations

• Generators are the basic elements that generate the entire free group through the group operation
  • Analogous to the concept of basis vectors in linear algebra
• Relations are equations or constraints that define how the generators interact with each other
  • In free groups, there are no relations other than the group axioms
• The set of generators is typically denoted by a set of symbols (letters, numbers)
• Each generator has an inverse element, which when combined with the generator using the group operation, yields the identity element
• Relations in a group presentation specify additional constraints or equations that the generators must satisfy
  • Example: Commutative relation $ab = ba$ for generators $a$ and $b$
• The number of generators in a free group determines its rank or the size of its generating set
• Free groups are characterized by having no relations other than the group axioms, allowing for the most general combination of generators

Free Group Construction and Properties

• Free groups can be constructed using words, which are finite sequences of generators and their inverses
  • Words represent the elements of the free group
• The group operation in a free group is concatenation of words, followed by simplification using the group axioms
  • Example: In a free group with generators $a$ and $b$, the words $ab$ and $ba$ are distinct elements
• Free groups satisfy the universal property, which states that any function from the set of generators to another group can be uniquely extended to a group homomorphism
• The word problem in free groups is solvable, meaning there is an algorithm to determine if two words represent the same element
• Free groups have a natural geometric interpretation as the fundamental group of a bouquet of circles, with each generator corresponding to a loop
• The subgroups of free groups are also free groups, with a possibly smaller rank
• Free groups are torsion-free, meaning they do not contain any non-trivial elements of finite order
• The rank of a free group is a well-defined invariant that characterizes its structure and complexity

Word Problems in Free Groups

• Word problems involve determining whether two words (sequences of generators and their inverses) represent the same element in the free group
• Solving word problems is crucial for understanding the structure and properties of free groups
• The word problem in free groups is decidable, meaning there exists an algorithm to determine if two words are equivalent
• Simplification techniques, such as cancellation of inverse pairs, are used to reduce words to their minimal form
  • Example: The word $aba^{-1}b^{-1}$ simplifies to the empty word (identity element)
• The solution to the word problem relies on the fact that each element in a free group has a unique reduced form (minimal representation)
• The word metric on a free group measures the distance between elements based on the length of the reduced words representing them
• Word problems can be visualized using van Kampen diagrams, which are planar diagrams that represent the equivalence of words in a group presentation
• Efficient algorithms, such as the Dehn algorithm, have been developed to solve word problems in free groups

Presentations: Defining Groups with Generators and Relations

• A group presentation is a way to define a group using a set of generators and a set of relations
  • Generators specify the basic elements that generate the group
  • Relations are equations that the generators must satisfy
• The presentation of a group is typically denoted as $\langle S \mid R \rangle$, where $S$ is the set of generators and $R$ is the set of relations
• The group defined by a presentation is the quotient of the free group on the generators by the normal subgroup generated by the relations
• Presentations provide a concise way to describe groups and their structure
  • Example: The cyclic group of order $n$ can be presented as $\langle a \mid a^n = 1 \rangle$
• Different presentations can define isomorphic groups, leading to the study of equivalent presentations
• The trivial group can be presented as $\langle \emptyset \mid \emptyset \rangle$, with an empty set of generators and relations
• Presentations can be used to study the properties and structure of groups, such as their subgroups, quotient groups, and homomorphisms
• The Tietze transformations provide a systematic way to manipulate and simplify group presentations while preserving the isomorphism type of the group

Tietze Transformations and Equivalent Presentations

• Tietze transformations are a set of operations that can be applied to a group presentation while preserving the isomorphism type of the group
• The four Tietze transformations are:
  1. Adding or removing a redundant generator (a generator that can be expressed in terms of the others)
  2. Adding or removing a redundant relation (a relation that follows from the other relations)
  3. Replacing a generator by a word in the other generators
  4. Replacing a relation by a conjugate relation
• Tietze transformations allow for the simplification and manipulation of group presentations
  • Example: The presentation $\langle a, b \mid ab = ba \rangle$ can be simplified to $\langle a \rangle$ by removing the redundant generator $b$
• Two presentations are called Tietze equivalent if one can be transformed into the other using a sequence of Tietze transformations
• Tietze equivalence is an equivalence relation on the set of group presentations
• The Tietze transformations provide a way to find simpler or more convenient presentations for a given group
• Tietze transformations can be used to prove that certain groups are isomorphic by finding a common presentation
• The Tietze transformations have applications in computational group theory and the study of group structures

Applications in Topology and Geometry

• Free groups have important applications in topology, particularly in the study of fundamental groups and covering spaces
• The fundamental group of a topological space is a free group if the space is a bouquet of circles or a graph
  • Each generator corresponds to a loop in the space
• Free groups are used to classify covering spaces of a topological space
  • The subgroups of the fundamental group correspond to the connected covering spaces
• The rank of the fundamental group (the number of generators) is a topological invariant that provides information about the structure of the space
• In algebraic topology, free groups are used to construct chain complexes and compute homology and cohomology groups
• Free groups also appear in the study of knot theory, where they are used to define knot invariants such as the knot group and the Alexander polynomial
• In geometric group theory, free groups serve as a model for understanding the geometry of groups and their actions on metric spaces
• The Cayley graph of a free group is a regular tree, which provides a geometric representation of the group structure
• Free groups are used in the study of hyperbolic geometry and the theory of hyperbolic groups, which have important applications in low-dimensional topology and geometric group theory

Computational Aspects and Algorithms

• Computational methods play a crucial role in the study of free groups and their applications
• The word problem in free groups can be solved efficiently using algorithms such as the Dehn algorithm or the Knuth-Bendix completion algorithm
  • These algorithms reduce words to their minimal form by applying cancellation and rewriting rules
• The conjugacy problem in free groups, which asks whether two elements are conjugate, can also be solved using algorithms based on the Dehn algorithm
• The membership problem, which asks whether a given word belongs to a subgroup of a free group, can be solved using the Stallings folding algorithm
  • This algorithm constructs a finite graph representation of the subgroup
• Algorithms for computing the rank and basis of a subgroup of a free group have been developed, such as the Nielsen-Schreier algorithm
• The Tietze transformation algorithms provide a way to computationally manipulate and simplify group presentations
• Algorithms for computing the automorphism group and the outer automorphism group of a free group have been studied, such as the Whitehead algorithm
• Computational tools and software packages, such as GAP (Groups, Algorithms, Programming) and MAGMA, provide implementations of various algorithms and methods for working with free groups
• The study of computational aspects of free groups has applications in cryptography, coding theory, and computer science, where free groups are used to model and analyze various algebraic and geometric structures