DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.

Polynomial, subexponential, exponential, and intermediate growth

Definition

Let $G$ be a finitely generated group, and let $S$ be any finite generating set. Write $β_{G} : = β_{G, S}$ .

By Growth type is independent of the finite generating set and the transitivity of $≼$ from Growth comparison is a preorder, the following conditions do not depend on the choice of $S$ .

$G$ has polynomial growth if $β_{G} ≼ n^{d}$ for some integer $d \geq 0$ .
$G$ has exponential growth if $a^{n} ≼ β_{G}$ for some natural number $a \geq 2$ .
$G$ has subexponential growth if it does not have exponential growth.
$G$ has intermediate growth if it has subexponential growth but not polynomial growth.

The comparison relation $≼$ is that of Growth comparison and growth type, and $β_{G, S}$ is from The growth function of a finitely generated group.

Depends on

Used by

Dependency tree · two levels

8 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)