DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-29

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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Algebraic relator area and the Dehn function of a finite presentation

Definition

Fix a finite presentation $P = ⟨ X ∣ R ⟩$ . If a word $w$ on $X ⊔ X^{- 1}$ is trivial in the presented group, then by The normal closure of $R$ is the set of finite products of conjugates of elements of $R$ and their inverses it can be written in the free group as

$w = \prod_{j = 1}^{m} u_{j} r_{j}^{ε_{j}} u_{j}^{- 1},$

where each $r_{j} \in R$ and each $ε_{j} \in {1, - 1}$ .

The algebraic relator area ${Area}_{P} (w)$ is the least such integer $m$ . The Dehn function of $P$ is

$δ_{P} (n) = \max {{Area}_{P} (w) : w =_{P} 1, ∣ w ∣ \leq n} .$

It is defined on $N$ and only measures null words.

Depends on

Used by

Dependency tree · two levels

8 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

John Meier, Groups, Graphs and Trees (standard reference, not scraped)
Dexter Chua after H. Wilton, Topics in Geometric Group Theory (standard reference, not scraped)