DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

How 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.

Growth comparison and growth type

Definition

Let $f, g : N \to N$ be nondecreasing functions.

Write $f ≼ g$ if there is a natural number $C \geq 1$ such that $f (n) \leq C g (C n + C) + C for every n \in N .$

Write $f ≃ g$ if both $f ≼ g$ and $g ≼ f$ hold.

For a finitely generated group $G$ and a finite generating set $S$ , the equivalence class of the growth function $β_{G, S}$ (The growth function of a finitely generated group) under $≃$ is the growth type of $G$ with respect to $S$ .

Depends on

The growth function of a finitely generated group

Used by

Dependency tree · two levels

4 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)
C. Drutu and M. Kapovich, Lectures on Geometric Group Theory, Chapter 5 (standard reference, not scraped)