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

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

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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A minimal-area van Kampen diagram is reduced

Statement

A van Kampen diagram of minimal area for its boundary word is reduced.

Facts & Assumptions

Given: A van Kampen diagram $D$ whose area is minimal among all diagrams with the same boundary label.

[L1]

In a reduced diagram there is no cancellable adjacent face pair (A reduced van Kampen diagram has no cancellable adjacent faces).

[F1]

The area of a van Kampen diagram is its number of $2$ -cells (Van Kampen diagrams, boundary labels, and diagram area for a presentation).

Proof

technique · direct

1.1

Suppose $D$ were not reduced. Then by [L1] there would be two adjacent 2-cells whose common edge can be cancelled. Delete those two faces and glue together the remaining boundary arcs.

L1givenconstruct

2.1

The surgery of step 1.1 does not change the outer boundary word, but it removes exactly two 2-cells. By [F1], the new diagram therefore has strictly smaller area than $D$ .

F1step 1.1algebra

3.1

This contradicts the assumed minimality of $D$ . Hence $D$ is reduced.

step 2.1contradiction: minimal area cannot drop∎

Depends on

Used by

Dependency tree · two levels

4 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)