ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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An ascending HNN extension from doubling the integers

Example

The injective endomorphism $ϕ : Z \to Z$ given by $ϕ (n) = 2 n$ produces the ascending HNN extension

$⟨ a, t ∣ t a t^{- 1} = a^{2} ⟩,$

and every element has a unique one-sided normal form $t^{- m} a^{k} t^{n}$ with $m, n \geq 0$ and $k$ odd whenever $m, n > 0$ .

Facts & Assumptions

Given: The doubling endomorphism of $Z$ .

[L1]

An injective endomorphism of a group defines an ascending HNN extension. (Ascending HNN extensions of injective endomorphisms)

[L2]

In an ascending HNN extension, every element has a unique form $t^{- m} b t^{n}$ , with $b$ outside the image subgroup whenever $m, n > 0$ . (Ascending HNN extensions admit the one-sided normal form)

Verification

technique · direct

1.1L1given

The map $ϕ (n) = 2 n$ is injective, so [L1] gives the presentation $⟨ a, t ∣ t a t^{- 1} = a^{2} ⟩$ . Its positive associated subgroup is $2 Z \leq Z$ .

2.1L2step 1.1algebra∎

Under the multiplicative notation $a^{k} \leftrightarrow k$ , the image subgroup $ϕ (Z) = 2 Z$ consists exactly of the even exponents. Thus $a^{k} \notin ϕ (Z)$ exactly when $k$ is odd. The condition in [L2] therefore specializes to the stated unique forms $t^{- m} a^{k} t^{n}$ , with $k$ odd whenever $m, n > 0$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

C. Loh, Geometric Group Theory: An Introduction (2015 course version) (standard reference, not scraped)