Free group on a set of generators

Depends on

• Group and abelian group
• Monoid homomorphism and group homomorphism
• Words in an alphabet with formal inverses, elementary cancellation, and reduced words

Used by

• Compute the twisted Burau determinant for a colored braid closure Algorithm
• A free product of copies of the infinite cyclic group is a free group Corollary
• A group acting freely without inversions on a tree is free Corollary
• The Artin action solves the braid word problem Corollary
• A conjugate-permuting automorphism that does not fix the boundary word is not in the braid image Counterexample
• Equal abelianizations do not classify knots Counterexample
• The free group on two generators is not amenable Counterexample
• The induced permutation does not determine a braid Counterexample
• A free basis of a group Definition
• An HNN extension with its stable letter Definition
• Arbitrary-rank Coxeter presentations, length, reflections, and Bruhat order Definition
• Artin automorphisms of the free group Definition
• Artin monoid and Artin group presentations, and the canonical monoid-to-group map Definition
• Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups Definition
• Free-presentation kernel data Definition
• Group presentation by generators and relations Definition
• Lower-central series and Magnus expansion for links Definition
• Peripheral-boundary-preserving automorphisms of Fₙ Definition
• The admissible Wada quotient of a deficiency-one exterior spine Definition
• The labeled Schreier coset graph of a subgroup of a free group Definition
• The total winding homomorphism of the punctured disk Definition
• Ping pong for two loxodromics Example
• The Cayley graph of the free group on two generators is the tree in which every vertex has four neighbours Example
• The free group on X represents G mapstoSet(X,U(G)) Example
• The free-group functor F:Set toGrp and free-module functor R⁽⁻⁾:Set→ R-Mod Example
• The full twist acts by boundary conjugation Example
• The two-circle wedge has both regular and nonregular connected three-sheeted coverings Example
• Worked Wirtinger presentations Example
• FALSE: any two infinite finitely generated groups are quasi-isometric False statement
• A based self-map of the punctured disk inducing the identity on the fundamental group is based-homotopic to the identity Lemma
• Artin automorphisms permute meridian conjugacy classes and fix the boundary word Lemma
• Artin's product-cancellation dichotomy Lemma
• Explicit unknot and figure-eight oriented Wirtinger codes Lemma
• Finite nilpotent link quotients are generated by component meridians Lemma
• The Artin automorphisms satisfy the braid relations Lemma
• The deficiency-one Fox calculus rule for the Alexander invariant Lemma
• The evaluated Fox Jacobian is the twisted cellular boundary Lemma
• The Schreier coset graph is connected and deterministic Lemma
• The standard flower is a deformation retract with free meridian basis Lemma
• The two cyclic basis factors of the rank-two free group are self-commensurating with trivial cross-conjugate intersections Lemma

…and 21 more results.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• McKernan, Presentations and Groups of Small Order, Lecture 12 (standard reference, not scraped)