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Under the ultrafilter lemma, subexponential growth implies amenability

Statement

Assume the ultrafilter lemma. Every finitely generated group of subexponential growth is amenable.

Facts & Assumptions

Given: A finitely generated group $G$ with a finite generating set $S$ , subexponential growth, and the ultrafilter lemma.

[L1]

The growth function counts word-metric balls (The growth function of a finitely generated group).

[L2]

Subexponential growth means that no exponential lower bound occurs (Polynomial, subexponential, exponential, and intermediate growth).

[L3]

Under the ultrafilter lemma, the Folner condition implies amenability (Under the ultrafilter lemma, the Folner condition is equivalent to amenability).

Proof

technique · direct

1.1

Let $B_{n}$ be the word-metric ball of radius $n$ about the identity. If some $δ > 0$ satisfied $∣ S B_{n} ∣ \geq (1 + δ) ∣ B_{n} ∣$ for every $n$ , then iterating would give $∣ B_{n} ∣ \geq (1 + δ)^{n}$ up to multiplicative constants, contradicting the subexponential alternative in [L2]. Therefore for every $ε > 0$ there exists $n$ with $∣ S B_{n} ∖ B_{n} ∣ < (ε / 2) ∣ B_{n} ∣$ .

L1L2givenalgebra

2.1

For such an $n$ , every $s \in S$ satisfies $∣ s B_{n} ∖ B_{n} ∣ \leq ∣ S B_{n} ∖ B_{n} ∣ < (ε / 2) ∣ B_{n} ∣$ , so the boundary formulation of [L3] makes $B_{n}$ an $(S, ε)$ -Folner set. Hence $G$ satisfies the Folner condition, and [L3] shows that $G$ is amenable.

L3step 1.1algebra∎

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Sources

C. Löh, Geometric Group Theory: An Introduction (2015 course version) (standard reference, not scraped)
Cornelia Drutu and Michael Kapovich, Lectures on Geometric Group Theory (standard reference, not scraped)