CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck 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.

Vertex groups embed in the fundamental group of a graph of groups

Statement

For every vertex $v$ of a graph of groups $G$ , the canonical map $G_{v} \to π_{1} (G, T)$ is injective.

Facts & Assumptions

Given: A graph of groups $G$ , a maximal subtree $T$ , and a vertex $v$ .

[L1]

Every element of the graph-of-groups fundamental group has a reduced normal form, and a reduced word of positive edge length is nonidentity. (Normal form for the fundamental group of a graph of groups)

Proof

technique · direct

1.1L1given

A nonidentity element of $G_{v}$ is already a reduced graph-of-groups word of edge length $0$ . The edge-length- $0$ clause of [L1] says that such a reduced word represents the identity only when its unique coefficient is the identity of $G_{v}$ .

2.1step 1.1algebra∎

Therefore no nonidentity element of $G_{v}$ maps to the identity in $π_{1} (G, T)$ . So the canonical map $G_{v} \to π_{1} (G, T)$ is injective.

Depends on

Normal form for the fundamental group of a graph of groups

Used by

The Bass-Serre tree of a graph of groups Definition
FALSE: the quotient graph determines the acting group without stabilizer data False statement
The fundamental group acts without inversions on its Bass-Serre tree Theorem

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

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