ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-09

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Hirsch length and growth degree differ

Example

The nilpotent Hirsch length and growth degree need not agree: for the integer Heisenberg group H, h(H)=3 and D(H)=4; for ${UT}_{4} (Z)$ , h=6 and D=10.

Facts & Assumptions

Given: Use the ranks computed in the two matrix examples.

[F1]

The Heisenberg lower-central ranks are 2,1 (The discrete Heisenberg group has growth degree four).

[F2]

The UT_4 lower-central ranks are 3,2,1 (UT_4(Z) has ranks three, two, one and growth degree ten).

[F3]

h sums ranks and D sums ranks multiplied by their layer (Bass–Guivarc’h dimension and nilpotent Hirsch length).

Verification

1.1

For H, the factor ranks (2,1) give $h = 2 + 1 = 3$ and $D = 1 \cdot 2 + 2 \cdot 1 = 4$ . Their difference is $(2 - 1) \cdot 1 = 1$ , contributed by the central second-layer free generator.

F1F3

2.1

For ${UT}_{4} (Z)$ the ranks (3,2,1) give $h = 3 + 2 + 1 = 6$ and $D = 1 \cdot 3 + 2 \cdot 2 + 3 \cdot 1 = 10$ . The difference is $(2 - 1) \cdot 2 + (3 - 1) \cdot 1 = 4$ . Thus h counts each free coordinate once, while D records its lower-central layer; these two explicit nilpotent groups have different values of the two invariants.

F2F3algebra∎

Source notes

Druţu–Kapovich, Geometric Group Theory (837-page edition), Definition 13.46, p.474. Revised Definition 13.46 supplies the distinction between the two sums; actual factor calculations are cited at their uses.

Depends on

Used by

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Dependency tree · two levels

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Sources

Druţu–Kapovich, Geometric Group Theory (837-page edition) (standard reference, not scraped)