TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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Collins' conjugacy theorem for cyclically reduced HNN words

Statement

Let $u$ and $v$ be cyclically Britton-reduced HNN words of positive stable-letter length. Then $u$ and $v$ are conjugate in the HNN extension if and only if some cyclic permutation $u^{'}$ of $u$ is conjugate to $v$ by an element of the base group.

In particular, conjugate cyclically Britton-reduced words of positive stable-letter length have the same stable-letter length. The length-zero base-group case of Collins' theorem is deferred: it requires a separate chain criterion through the associated subgroups.

Facts & Assumptions

Given: The cyclically Britton-reduced words $u$ and $v$ of positive stable-letter length.

[L1]

Every cyclic permutation of a positive-length cyclically Britton-reduced HNN word is conjugate to the original word and remains cyclically Britton-reduced. (Cyclic permutations of a cyclically Britton-reduced HNN word stay in the same conjugacy class)

[L2]

Conjugacy between positive-length cyclically Britton-reduced words reduces to conjugacy by a base-group element after a cyclic permutation. (Conjugacy between cyclically Britton-reduced HNN words reduces to base-group conjugacy after cyclic permutation)

Proof

technique · direct

1.1

Suppose first that $u$ and $v$ are conjugate. Then [L2] gives a cyclic permutation $u^{'}$ of $u$ that is conjugate to $v$ by an element of the base group. Because a cyclic permutation preserves the number of stable letters, conjugate positive-length cyclically Britton-reduced words have the same stable-letter length.

L2given

1.2

Conversely, if some cyclic permutation $u^{'}$ of $u$ is conjugate to $v$ by an element of the base group, then [L1] shows that $u$ is conjugate to $u^{'}$ . Composing with the given base-group conjugacy yields that $u$ is conjugate to $v$ .

L1given

2.1

The two directions prove the equivalence.

step 1.1step 1.2∎

Depends on

Used by

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Sources

Roger C. Lyndon and Paul E. Schupp, Combinatorial Group Theory (standard reference, not scraped)