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

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Reduced C prime(1/6) diagrams satisfy the standard combinatorial curvature count

Statement

Let $D$ be a reduced van Kampen diagram with at least one $2$ -cell over a symmetrised $C^{'} (1 / 6)$ presentation, and assume its outer boundary word is freely reduced and nontrivial. Then some boundary face of $D$ is a shell whose inner boundary is a concatenation of at most three maximal internal arcs.

Facts & Assumptions

Given: A reduced van Kampen diagram $D$ with at least one $2$ -cell over a symmetrised $C^{'} (1 / 6)$ presentation, with freely reduced nontrivial outer boundary word.

[L0]

A nondegenerate van Kampen diagram is a finite combinatorial $2$ -complex whose underlying space is a closed disc (Van Kampen diagrams, boundary labels, and diagram area for a presentation).

[F1]

The relator set satisfies the strict metric condition $C^{'} (1 / 6)$ (The small-cancellation conditions C(lambda) and C prime(lambda)).

[L1]

The diagram is reduced in the sense that no cancellable adjacent face pair occurs (A reduced van Kampen diagram has no cancellable adjacent faces).

[F2]

Under Section 3.5's standing $C^{'} (1 / 6)$ hypothesis, Touikan first observes that internal arcs of a reduced diagram are labelled by pieces and that every internal face of its arc reduction has at least seven sides; Definition 3.5.3 defines an $i$ -shell, and Proposition 3.5.5 states that an arc-reduced disc diagram contains an $i$ -shell for some $1 \leq i \leq 3$ . Independently, Abgrall--Munro Lemma 2.12 states the general Greendlinger form that every nontrivial reduced $C^{'} (1 / 6)$ disc diagram has $3$ -shells and/or boundary spurs.

Proof

technique · direct

1.1

If $D$ has exactly one $2$ -cell, that face has empty inner boundary and is therefore a shell with zero internal arcs. Hence assume that $D$ has at least two faces.

L0given

2.1

Collapse every maximal arc of $D$ ---boundary arcs as well as internal arcs---by suppressing its valence- $2$ internal vertices. The resulting diagram $\overline{D}$ is an arc-reduced combinatorial disc with the same faces and face incidences. Removing subdivisions neither creates a cancellable face pair nor changes which face-boundary portions are internal or external.

L0L1step 1.1construct

3.1

Every internal edge of $\overline{D}$ represents a maximal internal arc of $D$ . By reducedness, the two incident face occurrences do not cancel, so [F2] identifies the arc label as a piece. If $f$ is an interior face, these piece-arcs cover $\partial f$ ; [F1] makes each one shorter than $∣ \partial f ∣ / 6$ , so $f$ has at least seven sides. This is exactly the arc-reduced $C^{'} (1 / 6)$ setup preceding the proposition cited in [F2].

F1F2L1step 2.1algebra

4.1

Apply the Touikan proposition cited in [F2] to $\overline{D}$ . It gives a boundary face whose inner boundary consists of $i$ internal arcs for some $1 \leq i \leq 3$ . Expanding the suppressed valence- $2$ vertices turns those $i$ edges back into the same $i$ maximal internal arcs of $D$ , without changing the face or its outer boundary. Together with the one-face case in step 1.1, this proves the claim.

F2step 1.1step 2.1step 3.1∎

Depends on

Used by

In a reduced C prime(1/6) null diagram, some face contributes more than half of its boundary to the outer boundary Theorem

Dependency tree · two levels

6 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)
Adrien Abgrall and Zachary Munro, On residual finiteness of graphs of free groups with cyclic edge groups, Lemma 2.12 (standard reference, not scraped)