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

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Cyclic permutations of a cyclically Britton-reduced HNN word stay in the same conjugacy class

Statement

Let

$w = a_{0} t^{ε_{1}} a_{1} \dots t^{ε_{n}} a_{n}$

be cyclically Britton-reduced with $n > 0$ . Then every cyclic permutation of its stable-letter syllables is conjugate to $w$ . In particular,

$t^{ε_{1}} a_{1} \dots t^{ε_{n}} (a_{n} a_{0})$

lies in the conjugacy class of $w$ and is still cyclically Britton-reduced.

Facts & Assumptions

Given: The cyclically Britton-reduced word in the statement.

[L1]

A cyclically Britton-reduced word remains Britton-reduced after the end-rotation that moves the last stable-letter syllable to the front. (Cyclically Britton-reduced HNN words)

Proof

technique · direct

1.1

Conjugating $w$ by $a_{0}$ gives $a_{0}^{- 1} w a_{0} = t^{ε_{1}} a_{1} \dots t^{ε_{n}} (a_{n} a_{0})$ , so the displayed first cyclic permutation lies in the conjugacy class of $w$ .

givenalgebra

2.1

The end-rotation of the word from step 1.1 is $t^{ε_{n}} (a_{n} a_{0}) t^{ε_{1}} a_{1} \dots t^{ε_{n - 1}} a_{n - 1}$ , which is Britton-reduced by [L1]. That is exactly the criterion saying the word from step 1.1 is cyclically Britton-reduced. Repeating the same conjugation on each successive rotation shows that every cyclic permutation of the stable-letter syllables is conjugate to $w$ .

L1step 1.1algebra∎

Depends on

Cyclically Britton-reduced HNN words

Used by

Dependency tree · two levels

2 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

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