LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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 boundary monomorphisms from stabilizers are well-defined

Statement

In the quotient graph of groups attached to a tree action, changing the chosen lifts of a quotient edge and its endpoints conjugates the two boundary monomorphisms by the corresponding vertex-group identifications. Hence the construction is well defined up to canonical conjugacy.

Facts & Assumptions

Given: A group action without inversions on a tree and a chosen quotient edge $e$ .

[L1]

The quotient graph-of-groups construction uses stabilizers of chosen lifts, with the terminus map defined by a connecting element $g_{e}$ . (The quotient graph of groups attached to a tree action)

[L2]

Stabilizers of points in the same orbit are conjugate. (If $y = g \cdot x$ , then $G_{y} = g G_{x} g^{- 1}$ )

Proof

technique · direct

1.1L1L2given

If the chosen lift $\tilde{e}$ is replaced by $h \tilde{e}$ , then the edge stabilizer changes from $Stab (\tilde{e})$ to $h Stab (\tilde{e}) h^{- 1}$ by [L2], and the same conjugation change occurs for the endpoint stabilizers of $o (\tilde{e})$ and $t (\tilde{e})$ .

2.1L1step 1.1algebra

The new connecting element can be taken as $h g_{e} k^{- 1}$ when the terminal lift is changed by $k$ . Therefore the new terminus map sends $h x h^{- 1}$ to $k (g_{e}^{- 1} x g_{e}) k^{- 1}$ . This is exactly the old boundary monomorphism conjugated by the vertex-group identifications from step 1.1.

3.1step 2.1∎

Hence the boundary monomorphisms are independent of representative choices up to canonical conjugacy.

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

Jean-Pierre Serre, Trees (standard reference, not scraped)