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.
Gromov's polynomial-growth theorem
Gromov's polynomial-growth theorem states that a finitely generated group has polynomial growth if and only if it is virtually nilpotent.
This page does not prove that theorem. The forward implication is a deep structural result, not a consequence of the Švarc-Milnor and growth-comparison machinery developed here. The corollary Finitely generated nilpotent groups have polynomial growth supplies the nilpotent case of the backward implication; passing from nilpotent to virtually nilpotent also requires the finite-index quasi-isometry argument.
Depends on
Used by
- FALSE: Gromov's polynomial-growth theorem is proved on this page False statement
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
- C. Löh, Geometric Group Theory, Sections 5.1-5.3 (standard reference, not scraped)
- M. Gromov, Groups of polynomial growth and expanding maps (standard reference, not scraped)