TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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Free groups of rank at least two have exponential growth

Statement

Let $F_{r}$ be a free group of rank $r \geq 2$ . Then $F_{r}$ has exponential growth.

Facts & Assumptions

Given: A free group $F_{r}$ of rank $r \geq 2$ together with a free basis $X$ of size $r$ .

[L1]

In the word metric defined by a free basis, word length is exactly reduced-word length (With respect to a free basis, the word length of an element is the length of its reduced word).

[L2]

The growth function counts elements with bounded word length (The growth function of a finitely generated group).

[L3]

Exponential growth means that $a^{n} ≼ β_{G}$ for some real $a > 1$ (Polynomial, subexponential, exponential, and intermediate growth).

[L4]

A free group of rank $r$ has a free basis with $r$ elements (The rank of a free group admitting a finite basis).

[L5]

Reduced words form a free group on the basis alphabet, and any two free groups on that alphabet are uniquely isomorphic compatibly with their generators; hence distinct reduced words represent distinct elements of $F_{r}$ (Reduced words form the free group on an alphabet, Free groups on the same set are uniquely isomorphic compatibly with their generators).

Proof

technique · direct

1.1

For each $n \geq 1$ , the reduced words of length exactly $n$ on $X \cup X^{- 1}$ number $2 r (2 r - 1)^{n - 1}$ : there are $2 r$ choices for the first letter and, after that, $2 r - 1$ choices at each step to avoid immediate cancellation.

L4algebra

2.1

By [L1] and [L5], those reduced words represent distinct elements of word length exactly $n$ . Therefore the ball of radius $n$ contains at least $2 r (2 r - 1)^{n - 1}$ elements, so $β_{F_{r}, X} (n) \geq (2 r - 1)^{n}$ for every $n \geq 1$ .

L1L2L5step 1.1algebra

3.1

Because $r \geq 2$ , the real number $a : = 2 r - 1$ satisfies $a > 1$ . Step 2.1 gives $a^{n} \leq β_{F_{r}, X} (n)$ for all $n$ , so $a^{n} ≼ β_{F_{r}, X}$ and [L3] makes the growth exponential.

L3step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

23 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)