Alphabeta MathHow 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.
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Bass-Guivarch growth-degree formula
For a finitely generated nilpotent group , let be the homogeneous dimension from The homogeneous dimension of a finitely generated nilpotent group. The Bass-Guivarch growth-degree formula says that, for every finite generating set ,
This page does not prove that formula or the accompanying structural fact used to state it. The source-backed result records that the lower-central quotients are finitely generated abelian, so their displayed free ranks and the sum defining are well defined.
Depends on
Used by
- Finitely generated nilpotent groups have polynomial growth Corollary
- The discrete Heisenberg group has growth degree four Example
Cited to discharge well-definedness by The homogeneous dimension of a finitely generated nilpotent group.
Dependency tree · two levels
10 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)
- H. Bass, The degree of polynomial growth of finitely generated nilpotent groups (standard reference, not scraped)