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

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The growth function of a finitely generated group

Definition

Let $G$ be a finitely generated group, and let $S$ be a finite generating set (Finitely generated groups).

The growth function of $G$ with respect to $S$ is $β_{G, S} : N \to N, β_{G, S} (n) : = ∣ {g \in G : ∣ g ∣_{S} \leq n} ∣ .$

Since $d_{S} (e, g) = ∣ g ∣_{S}$ (The word metric of a group with respect to a generating set, Word length of a group element with respect to a generating set), this is exactly the cardinality of the closed word-metric ball of radius $n$ about the identity.

Depends on

Used by

Growth comparison and growth type Definition
Polynomial, subexponential, exponential, and intermediate growth Definition
FALSE: the growth function is independent of the generating set pointwise False statement
Bass-Guivarch growth-degree formula Remark
Free groups of rank at least two have exponential growth Theorem
Growth type is a quasi-isometry invariant of finitely generated groups Theorem
Growth type is independent of the finite generating set Theorem

Dependency tree · two levels

11 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)