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.
A product of two infinite groups need not be hyperbolic
Statement refuted
The product of two infinite groups need not be hyperbolic.
Facts & Assumptions
Given: The direct product .
Free abelian groups of rank at least two are not hyperbolic (Free abelian groups of rank at least two are not hyperbolic).
Counterexample
The group is a direct product of two infinite groups and is free abelian of rank .
Therefore [L1] shows that is not hyperbolic. So the general claim fails.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
5 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
- Clara Löh, Geometric Group Theory, Section 6.5.4 (standard reference, not scraped)