False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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FALSE: every repeated subword of a relator is a piece

Statement

Every repeated subword of a relator is a piece.

Facts & Assumptions

Given: The one-relator symmetrised set generated by $r = a b a b$ .

[L1]

A piece must occur as an initial segment in two distinct symmetrised occurrences (A piece is a common initial segment occurring in two distinct places of a symmetrised relator set).

Refutation

technique · direct

1.1

The reduced subword $a b$ appears twice inside $r$ , namely in positions $1$ through $2$ and $3$ through $4$ . So it is certainly a repeated subword of one relator.

given

2.1

The cyclic conjugates of $r$ are only $a b a b$ and $b a b a$ , while the cyclic conjugates of $r^{- 1}$ start with inverse letters. Among these symmetrised occurrences, $a b$ is an initial segment only of $a b a b$ , and in that word the continuation is always the same suffix $a b$ . Therefore there is no second distinct ordered pair $(s, v)$ with $s = a b v$ , so [L1] says that $a b$ is not a piece.

L1step 1.1

3.1

So a repeated interior subword need not be a piece. The statement is false.

step 1.1step 2.1∎

Depends on

A piece is a common initial segment occurring in two distinct places of a symmetrised relator set

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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)