TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Growth type is independent of the finite generating set

Statement

Let $G$ be a finitely generated group, and let $S$ and $T$ be finite generating sets. Then the growth functions $β_{G, S}$ and $β_{G, T}$ are equivalent under $≃$ . Hence the growth type of a finitely generated group does not depend on the chosen finite generating set.

Facts & Assumptions

Given: A finitely generated group $G$ and finite generating sets $S$ and $T$ .

[L1]

The growth function $β_{G, S} (n)$ counts the elements with $∣ g ∣_{S} \leq n$ , and $β_{G, T} (n)$ is defined similarly (The growth function of a finitely generated group).

[L2]

The relation $≃$ is the mutual comparison relation generated by $f (n) \leq C g (C n + C) + C$ for some natural number $C \geq 1$ (Growth comparison and growth type).

[L3]

The identity map between the two word metrics is a bilipschitz equivalence (The identity map between the word metrics of two finite generating sets is a bilipschitz equivalence).

Proof

technique · direct

1.1L3choose

By [L3], choose a natural number $C \geq 1$ such that $d_{T} (g, h) \leq C d_{S} (g, h)$ and $d_{S} (g, h) \leq C d_{T} (g, h)$ for all $g, h \in G$ .

2.1L1step 1.1

If $∣ g ∣_{S} \leq n$ , then step 1.1 gives $∣ g ∣_{T} \leq C n$ . So every element counted by $β_{G, S} (n)$ is also counted by $β_{G, T} (C n)$ , and therefore $β_{G, S} (n) \leq β_{G, T} (C n)$ . Exchanging $S$ and $T$ yields the reverse inequality.

3.1L2step 2.1∎

Because $C \geq 1$ and the growth functions are nondecreasing, step 2.1 implies $β_{G, S} (n) \leq β_{G, T} (C n) \leq C β_{G, T} (C n + C) + C,$ and likewise with $S$ and $T$ interchanged. Thus [L2] gives $β_{G, S} ≼ β_{G, T}$ and $β_{G, T} ≼ β_{G, S}$ . Hence $β_{G, S} ≃ β_{G, T}$ , and the growth type is independent of the finite generating set.

Depends on

Used by

Polynomial, subexponential, exponential, and intermediate growth Definition
Free abelian groups have polynomial growth of the expected degree Example
The discrete Heisenberg group has growth degree four Example
FALSE: the growth function is independent of the generating set pointwise False statement

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)