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LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28
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Every HNN conjugacy class contains a cyclically Britton-reduced representative

Statement

Every element of an HNN extension is conjugate to a cyclically Britton-reduced HNN word.

Facts & Assumptions

Given: An element of an HNN extension.

[L1]

A cyclically Britton-reduced word is a Britton-reduced word with no pin across its two ends. (Cyclically Britton-reduced HNN words)

[L2]

Elementary pin reductions preserve the represented element. (Elementary HNN reductions preserve the represented element)

[L3]

A Britton-reduced word containing a stable letter is nontrivial. (Britton's lemma)

Proof

technique · direct
1.1

Choose, among all conjugates of the given element, a Britton-reduced representative w of minimal stable-letter length. Such a representative exists because one may first Britton-reduce any conjugate using [L2].

L2givenchoose
2.1

If w were not cyclically Britton-reduced, [L1] would give a pin across the two ends. Conjugating by the initial stable-letter syllable rotates that end-pin into the interior of the word, and [L2] then removes it to produce a conjugate with strictly smaller stable-letter length, contradicting the minimal choice in step 1.1.

L1L2step 1.1algebra
3.1

Therefore the minimal Britton-reduced representative from step 1.1 has no end-pin and is cyclically Britton-reduced. The nontriviality clause [L3] ensures that the shortening in step 2.1 is genuine whenever a stable letter is present.

L1L3step 2.1∎

Depends on

Used by

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