Group presentation by generators and relations

Depends on

• Free group on a set of generators
• The normal closure of a subset of a group
• The quotient group $G/N$ and coset product $(gN)(hN)=ghN$

Used by

• Every finite group has a finite presentation from its multiplication table Corollary
• In ⟨ a,b∣ ab, aba⟩, delete-only relator rewriting sends aba either to the empty word or to the stuck word a Counterexample
• In ⟨ a,b∣ ab⟩, the trivial word ba is stuck under free cancellation and delete-only relator rewriting Counterexample
• In ⟨ a,b∣ aba⁻¹b⁻¹⟩, the trivial word a²b²a⁻²b⁻² is stuck under free cancellation and delete-only relator rewriting Counterexample
• An HNN extension with its stable letter Definition
• Recursive presentations and finite presentations of groups Definition
• Relators and relations; finitely generated, finitely related, and finite presentations Definition
• Sc toolkit symmetrised relators and pieces Definition
• Singular planar labelled relator diagrams and their outer walks Definition
• The braid group by Artin presentation Definition
• The free-presentation Lyndon bar bicomplex Definition
• The path group of a graph of groups Definition
• The symmetrisation of a relator set closes under inverses and cyclic conjugates Definition
• Tietze transformations: dictionary generators, redundant relators, renaming, and their inverses Definition
• Van Kampen diagrams, boundary labels, and diagram area for a presentation Definition
• ⟨ a,b∣ a², b², aba⁻¹b⁻¹⟩≅(ℤ/2)×(ℤ/2) Example
• ⟨ a,b∣ aba⁻¹b⁻¹⟩≅(ℤ,+)×(ℤ,+) Example
• ⟨ a∣ aⁿ⟩≅(ℤ/n,+) for every n≥ 1 Example
• A Baumslag-Solitar group gives a noninjective completion map Example
• A strict C prime(1/6) presentation that is a free group of rank six Example
• Amalgamating infinite cyclic groups by multiplication by m and n gives the presentation with relation xᵐ=yⁿ Example
• Dₙ≅⟨ r,s∣ rⁿ, s², srs⁻¹r⟩ for the dihedral group Dₙ=⟨{ρ,σ}⟩leqSym(ℤ/n), n≥ 3 Example
• Sym({0,1,2})≅⟨ s,t∣ s², t², (st)³⟩ Example
• The infinite dihedral group is quasi-isometric to ℤ, and to ℤ×ℤ/2 Example
• FALSE: Greendlinger's lemma holds for every finite presentation False statement
• Algebraic relator area controls coarse filling area Lemma
• Linear isoperimetry implies uniformly thin geodesic bigons Lemma
• Relator expressions admit singular planar diagrams with controlled incidence Lemma
• Short loop relators give a finite dehn presentation Lemma
• The boundary label of a van Kampen diagram is trivial in the presented group Lemma
• In ⟨ X∣ R⟩, the words u and v represent the same element if and only if u⁻¹v∈⟨⟨ R⟩⟩ Proposition
• A free product has the union presentation of presentations of its factors Theorem
• A free product with amalgamation has the factor presentations plus the amalgamating relations Theorem
• Every group admits a presentation Theorem
• Hyperbolic groups admit finite Dehn presentations Theorem
• Linear isoperimetric characterisation of hyperbolic groups Theorem
• The Reidemeister-Schreier presentation theorem Theorem
• Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group Theorem

Dependency tree · two levels

10 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

• Brittenham, Group Presentations, Class Notes (standard reference, not scraped)