ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The Bass-Serre tree of a Baumslag-Solitar group

Example

For nonzero integers $m, n$ , the Baumslag-Solitar group

$B S (m, n) = ⟨ a, t ∣ t a^{m} t^{- 1} = a^{n} ⟩$

is the fundamental group of a one-loop graph of groups with vertex group $Z$ and edge group $Z$ , where the two boundary maps are $k \mapsto m k$ and $k \mapsto n k$ .

Facts & Assumptions

Given: Nonzero integers $m, n$ and the one-loop graph-of-groups description of HNN extensions.

[L1]

A one-loop graph of groups yields the corresponding HNN extension. (A one-loop graph of groups gives an HNN extension)

[L2]

The Bass-Serre tree uses cosets of the vertex and edge groups. (The Bass-Serre tree of a graph of groups)

Verification

technique · direct

1.1

Because $m, n \neq 0$ , the maps $Z \to Z$ , $k \mapsto m k$ and $k \mapsto n k$ , are injective. Applying [L1] to these two monomorphisms gives exactly the presentation of $B S (m, n)$ .

L1givenalgebra

2.1

Therefore [L2] describes its Bass-Serre tree: vertices are left cosets of the vertex copy of $Z$ , edges are left cosets of the edge copy of $Z$ , and the stable letter translates between adjacent levels of this tree.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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

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