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CorollaryStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26‡ rests on unproved material (inherited)
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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Finitely generated nilpotent groups have polynomial growth

Statement

Every finitely generated nilpotent group has polynomial growth.

Facts & Assumptions

Given: A finitely generated nilpotent group G.

[A1]

Bass-Guivarch says that βG,S(n)≃nD(G) for every finite generating set S.

[L2]

Polynomial growth means that βG≼nd for some integer d≥0 (Polynomial, subexponential, exponential, and intermediate growth).

Proof

technique · direct
1.1

By [A1], the growth function of G is equivalent to the polynomial nD(G). In particular it is bounded above, in the growth-comparison sense, by a polynomial.

A1
2.1

Therefore [L2] makes G a group of polynomial growth.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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