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 Bass-Serre tree of a free product
Example
For a free product , the Bass-Serre tree has vertex set and one edge for each left coset of the trivial amalgamating subgroup.
Facts & Assumptions
Given: The Bass-Serre construction for a graph of groups.
The Bass-Serre tree uses cosets of vertex groups for vertices and cosets of edge groups for edges. (The Bass-Serre tree of a graph of groups)
The fundamental group acts on that tree with quotient the underlying graph. (The fundamental group acts without inversions on its Bass-Serre tree)
Verification
View as the graph of groups with two vertices, vertex groups and , and one trivial edge group between them. Then [L1] gives vertex cosets and , while the edge cosets are just the elements of itself.
By [L2], the quotient is a single segment. Thus each edge joins one -coset to one -coset, giving the usual bipartite Bass-Serre tree of the free product.
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)