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.
FALSE: a quotient of a tree by a group action is always a tree
Statement
Whenever a group acts on a simplicial tree without inversions, the quotient graph is again a tree.
Facts & Assumptions
Given: The quotient-graph definition and a hyperbolic tree automorphism.
A quotient graph keeps only vertex and edge orbits. (The quotient graph of an action without inversions)
Hyperbolic automorphisms act by translation on an invariant axis. (Tree automorphisms without inversions are either elliptic or hyperbolic)
Refutation
Let be translation by on the bi-infinite line. By [L2] this is a hyperbolic action without inversions on a tree.
In the quotient graph from [L1], the vertex orbits are the residue classes of modulo , and the edge orbits join them in a -cycle. That quotient is not a tree, so the statement is false.
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
- Yuriy Tumarkin, Groups Acting on Trees (standard reference, not scraped)