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Britton's lemma

Statement

Let

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

be a Britton-reduced HNN word. If $w$ represents the identity, then $n = 0$ and $a_{0} = e_{A}$ . Equivalently, every Britton-reduced HNN word containing a stable letter is nontrivial.

Facts & Assumptions

Given: The Britton-reduced word in the statement.

[L1]

A Britton-reduced word has no pin, so a change of sign can occur only across a coefficient outside the subgroup that would create a pin. (HNN words, pins, and Britton-reduced words)

[L2]

Relative to chosen transversals, every element has a unique transversal normal form, and the identity has the unique normal form of stable-letter length zero with trivial base coefficient. (Normal forms in an HNN extension are unique relative to chosen transversals)

Proof

technique · direct

1.1

Choose transversals containing the identity in both associated subgroups. Normalize $w$ by the procedure of [L2]. Because $w$ is Britton-reduced by [L1], no elementary pin reduction is available, so the normalization only replaces each interior coefficient by the corresponding transversal representative in the same coset and leaves the stable-letter length $n$ unchanged.

L1L2givenalgebra

2.1

If $w = 1$ , the resulting normal form is the identity's normal form from [L2]. Step 1.1 shows that this is possible only when $n = 0$ , and then uniqueness in [L2] forces the remaining coefficient to be $a_{0} = e_{A}$ . The contrapositive is exactly the nontriviality clause for Britton-reduced words containing a stable letter.

L2step 1.1algebra∎

Depends on

Used by

The base group embeds in its HNN extension Corollary
The stable letter has infinite order Corollary
Britton reduction of a word with two pins Example
FALSE: every word containing a stable letter is nontrivial False statement
Every HNN conjugacy class contains a cyclically Britton-reduced representative Lemma

Dependency tree · two levels

7 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)