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Under the ultrafilter lemma, subgroups and quotients of amenable groups are amenable

Statement

Assume the ultrafilter lemma. Every subgroup of an amenable group is amenable, and every quotient of an amenable group by a normal subgroup is amenable.

Facts & Assumptions

Given: An amenable group $G$ and the ultrafilter lemma.

[L1]

Amenability means existence of a left-invariant mean (Left-invariant means and amenable groups).

[L2]

Normal subgroups are the conjugation-invariant subgroups for which the quotient group is formed (Normal subgroup: invariance under conjugation).

[L3]

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

Proof

technique · direct

1.1

Let $H \leq G$ . To show that $H$ is amenable, by [L3] it is enough to verify the Folner condition in $H$ . Fix a finite subset $S \subseteq H$ and $ε > 0$ . If $S = \emptyset$ , then ${e} \subseteq H$ is already an $(S, ε)$ -Folner set. Assume now that $S \neq \emptyset$ , and put $δ = ε / ∣ S ∣$ . Since $G$ is amenable, [L3] gives a finite nonempty set $F \subseteq G$ with $∣ s F △ F ∣ < δ ∣ F ∣$ for every $s \in S$ . Write $F = ⨆_{j = 1}^{m} E_{j} t_{j}$ , where the $E_{j}$ are the nonempty intersections of $F$ with the finitely many right $H$ -cosets that meet $F$ , transported back into $H$ . For each $s \in S$ , left translation by $s$ preserves every right $H$ -coset $H t_{j}$ , so $s F △ F = ⨆_{j = 1}^{m} ((s E_{j}) t_{j} △ E_{j} t_{j})$ and hence $\sum_{j = 1}^{m} ∣ s E_{j} △ E_{j} ∣ = ∣ s F △ F ∣$ . Summing over $s \in S$ gives $\sum_{j = 1}^{m} \sum_{s \in S} ∣ s E_{j} △ E_{j} ∣ < ∣ S ∣ δ ∣ F ∣ = ε ∣ F ∣ = ε \sum_{j = 1}^{m} ∣ E_{j} ∣ .$ Therefore some $j$ satisfies $\sum_{s \in S} ∣ s E_{j} △ E_{j} ∣ < ε ∣ E_{j} ∣$ , and then each summand is itself $< ε ∣ E_{j} ∣$ . So $E_{j}$ is an $(S, ε)$ -Folner set in $H$ . Since $S$ and $ε$ were arbitrary, $H$ satisfies the Folner condition, and [L3] makes $H$ amenable.

L3givenalgebra

1.2

Let $N ⊴ G$ , and let $q : G \to G / N$ be the quotient map from [L2]. For bounded $u : G / N \to R$ , define $m_{G / N} (u) = m (u \circ q)$ using a left-invariant mean $m$ on $G$ . Then $m_{G / N}$ is a mean, and for $\overset{ˉ}{g} = g N$ one has $(\overset{ˉ}{g} \cdot u) \circ q = g \cdot (u \circ q)$ , so left invariance of $m$ implies $m_{G / N} (\overset{ˉ}{g} \cdot u) = m_{G / N} (u)$ . Therefore the quotient is amenable.

L1L2given

2.1

Steps 1.1 and 1.2 prove the two permanence statements.

step 1.1step 1.2∎

Depends on

Used by

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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)