DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

A graph of groups

Definition

A graph of groups $G$ consists of:

a connected oriented graph $X = (V, E)$ in the sense of An oriented graph with edge reversal,
a group $G_{v}$ for each vertex $v \in V$ ,
a group $G_{e} = G_{\overset{ˉ}{e}}$ for each geometric edge, and
for each oriented edge $e$ , an injective homomorphism $α_{e} : G_{e} ↪ G_{o (e)}$ .

The opposite orientation $\overset{ˉ}{e}$ carries the corresponding map $α_{\overset{ˉ}{e}} : G_{e} ↪ G_{t (e)}$ .

Depends on

Used by

Reduced words in a graph of groups Definition
The path group of a graph of groups Definition

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