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The base group embeds in its HNN extension
Statement
In an HNN extension, the canonical map from the base group to the presented group is injective. Equivalently, a base-group element represents the identity in the HNN extension only when it is already the identity in .
Facts & Assumptions
Given: An HNN extension of a base group .
If a Britton-reduced word represents the identity, then it has stable-letter length zero and trivial base coefficient. (Britton's lemma)
Proof
Regard as the HNN word of stable-letter length zero. It is Britton-reduced, since it contains no stable letters and hence no pin.
If this word represents the identity in the HNN extension, [L1] forces its unique base coefficient to be . Therefore distinct elements of remain distinct in the HNN extension, so the canonical map is injective.
Depends on
Used by
- FALSE: the base group may collapse in an HNN extension False statement
Dependency tree · two levels
3 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)