Alphabeta MathHow 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 fundamental group of a graph of groups relative to a maximal tree
Definition
Let be a graph of groups on a connected graph , and let be a maximal subtree. The fundamental group is the quotient of the path group by the additional relations
So the tree edges are collapsed, while the non-tree edges remain as stable letters.
Depends on
Used by
- A group acting freely without inversions on a tree is free Corollary
- Reduced words in a graph of groups Definition
- The Bass-Serre tree of a graph of groups Definition
- FALSE: the fundamental group of a graph of groups is a topological fundamental group by definition False statement
- A one-loop graph of groups gives an HNN extension Theorem
- A one-segment graph of groups gives an amalgamated free product Theorem
- Bass-Serre structure theorem Theorem
- Kurosh subgroup theorem Theorem
- Normal form for the fundamental group of a graph of groups 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)