TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29

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A one-loop graph of groups gives an HNN extension

Statement

If a graph of groups has one vertex and one loop edge outside the chosen maximal subtree, then its fundamental group is the HNN extension of the vertex group with associated subgroups the two edge-group images.

Facts & Assumptions

Given: A one-loop graph of groups with vertex group $A$ , edge group $C$ , and boundary maps $α, β : C ↪ A$ .

[L1]

An HNN extension adjoins one stable letter $t$ satisfying $t α (c) t^{- 1} = β (c)$ for every $c \in C$ . (An HNN extension with its stable letter)

[L2]

The relative fundamental group is obtained from the path group by killing the chosen tree edges; here the loop edge is not killed. (The fundamental group of a graph of groups relative to a maximal tree)

Proof

technique · direct

1.1

Since the quotient graph has one vertex and the loop edge is outside the maximal subtree, [L2] leaves one edge symbol $t$ together with the vertex group $A$ . The defining relation of the path group is exactly $t α (c) t^{- 1} = β (c)$ .

L2given

2.1

Therefore the resulting fundamental group is precisely the HNN extension described in [L1].

L1step 1.1∎

Depends on

Used by

The Bass-Serre tree of a Baumslag-Solitar group Example
FALSE: the quotient graph determines the acting group without stabilizer data False statement

Dependency tree · two levels

15 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)
C. Loh, Geometric Group Theory: An Introduction (2015 course version) (standard reference, not scraped)