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

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The homogeneous dimension of a finitely generated nilpotent group

Definition

Let $G$ be a finitely generated nilpotent group of class $c$ , and write $γ_{1} (G) = G, γ_{i + 1} (G) = [G, γ_{i} (G)]$ for its lower central series (Subgroup commutators and the lower central series, Nilpotent groups and nilpotency class).

The later source-backed remark Bass-Guivarch growth-degree formula ‡ records that each quotient $γ_{i} (G) / γ_{i + 1} (G)$ is a finitely generated abelian group, so its free rank as a $Z$ -module is defined (The free rank of a finitely generated module over a PID).

The homogeneous dimension of $G$ is $D (G) : = \sum_{i = 1}^{c} i \cdot {rank}_{Z} ⁣ (γ_{i} (G) / γ_{i + 1} (G)) .$

Depends on

Used by

The discrete Heisenberg group has growth degree four Example
Bass-Guivarch growth-degree formula Remark

Dependency tree · two levels

9 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)
J. Milnor, Growth of finitely generated solvable groups (standard reference, not scraped)