DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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The condition T(q) forbids short cycles of pieces in the relator graph

Definition

Let $q \geq 3$ be an integer and let $R^{sym}$ be a symmetrised relator set (The symmetrisation of a relator set closes under inverses and cyclic conjugates). The set $R^{sym}$ satisfies $T (q)$ when the following holds: whenever $r_{1}, \dots, r_{h} \in R^{sym}$ with $3 \leq h < q$ and no adjacent cyclic pair $r_{i}, r_{i + 1}$ is inverse to one another, at least one cyclic product $r_{i} r_{i + 1}$ is freely reduced as written. Here indices are read modulo $h$ : $r_{h + 1} : = r_{1}$ , and both clauses range over $i \in {1, \dots, h}$ .

So $T (q)$ rules out short cyclic chains of relators in which every neighbour pair cancels. It is the piece-cycle condition complementary to the metric $C$ -conditions.

Depends on

The symmetrisation of a relator set closes under inverses and cyclic conjugates

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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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)