TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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A word is trivial in a presented group exactly when it bounds a finite van Kampen diagram

Statement

Let $G = ⟨ X ∣ R ⟩$ and let $w$ be a word on $X^{\pm 1}$ . Then $w$ represents the identity in $G$ if and only if $w$ is the boundary label of a finite van Kampen diagram over $⟨ X ∣ R ⟩$ .

Facts & Assumptions

Given: A presentation $G = ⟨ X ∣ R ⟩$ and a word $w$ on $X^{\pm 1}$ .

[L1]

The boundary label of every van Kampen diagram is trivial in the presented group (The boundary label of a van Kampen diagram is trivial in the presented group).

[F1]

A word lies in the normal closure of $R$ exactly when it is a finite product of conjugates of relators and their inverses (The normal closure of $R$ is the set of finite products of conjugates of elements of $R$ and their inverses).

Proof

technique · direct

1.1

If $w$ is the boundary label of a finite van Kampen diagram, then [L1] says that $w$ represents the identity in $G$ .

L1given

1.2

Conversely, suppose that $w$ represents the identity in $G$ . Then $w$ lies in the normal closure of $R$ , so [F1] gives a factorisation $w = \prod_{k = 1}^{m} u_{k} r_{k}^{ε_{k}} u_{k}^{- 1}$ with $r_{k} \in R$ and $ε_{k} \in {\pm 1}$ .

F1given

2.1

For each factor $u_{k} r_{k}^{ε_{k}} u_{k}^{- 1}$ , take one 2-cell with boundary word $r_{k}^{ε_{k}}$ and attach to its boundary a whisker labelled $u_{k}$ from a common basepoint. Gluing these $m$ discs along the whiskers produces a finite planar diagram whose outer boundary label is exactly the product in step 1.2, namely $w$ .

step 1.2construct

3.1

Steps 1.1 and 2.1 prove both directions of the equivalence.

step 1.1step 2.1∎

Depends on

Used by

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

GAP SmallCancellation manual, Chapter 1: Small Cancellation Theory — the classical conditions (standard reference, not scraped)
Jay Williams, Universal Countable Borel Quasi-Orders (standard reference, not scraped)
Nicholas Touikan, An Introduction to Combinatorial and Geometric Group Theory, Section 3.5 (standard reference, not scraped)
Clara Löh, Geometric Group Theory: An Introduction, Section 7.4.1 (standard reference, not scraped)