Some closure properties of amenable groups

8 January 2024

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Here is a small collection of properties that amenable groups enjoy.

Definition: A discrete group $G$ is called amenable, if there exists a function $\mu\colon 2^G \to [0,1]$ such that

1. $\mu(G) = 1$.
2. For all $A,B \subseteq G$ with $A \cap B = \varnothing$ we have $\mu(A \cup B) = \mu(A) + \mu(B)$.
3. For all $A \subseteq G$ and $g \in G$ we have $\mu(Ag) = \mu(A)$.

Hence, $\mu$ is a finitely additive right-invariant probability measure on $G$. We call such a function $\mu$ an invariant mean of $G$.

Remark: Since $\mu$ is not $\sigma$-additive, we cannot directly apply the results of measure theory here. So first we have to convince ourselves, that we can integrate at least bounded functions with respect to invariant means.

Definition: A simple function on $G$ is a function $f\colon G \to \bR$ which is a linear combination of indicator functions

$$f(x) = \sum_{i=1}^n \lambda_i \mathbf{1}_{A_i}(x)$$

where $n \in \bN, ~ \lambda_i \in \bR, ~ A_i \subseteq G$. The simple functions form a vector space $\mathcal{E}(G)$.

Remark: From measure theory we know that simple functions have a standard form in which the indicators are of disjoint sets. Furthermore, every simple function has a finite range and is therefore bounded. The integral on simple functions can be defined exactly as for $\sigma$-additive measures.

Proposition: Let $(B(G), \norm{\cdot}_\infty)$ be the Banach space of bounded functions on $G$ equipped with the sup norm. The subspace $\mathcal{E}(G) \leq B(G)$ is dense.

Proof: Let $f\colon G \to \bR$ be a bounded function. Suppose $f$ is positive. Then $f(G) \subseteq [0, b)$ for some $b > 0$. Partition $[0, b)$ into $n$ subintervals $[a_i, b_i)$. Then the $A_i \coloneqq f^{-1}([a_i, b_i))$ form a partition of $G$. Now the function $g = \sum_{i=1}^n a_i \mathbf{1}_{A_i} \in \mathcal{E}(G)$ satisfies $\norm{f - g}_\infty \leq \eps$ where $\eps > 0$ is the mesh size of the partition. For the general case, split $f = f^+ - f^-$ and apply the same argument to $f^+$ and $f^-$. By refining the partition we can make the mesh size as small as needed, hence $\mathcal{E}(G)$ is dense in $B(G)$.

Corollary: The integral of bounded functions with respect to invariant means is well-defined.

Proof: Let $\mu$ be an invariant mean of $G$. Notice that integration with respect to $\mu$ defines a linear map

$$\mathcal{E}(G) \to \bR, \quad f \mapsto \int_G f \ d\mu$$

which is Lipschitz. Indeed, we have

$$\abs{\int_G f \ d\mu} \leq \int_G \abs{f} \ d \mu \leq \norm{f}_\infty \int_G \mathbf{1}_G \ d\mu = \norm{f}_\infty.$$

Since $\mathcal{E}(G)$ is dense in $B(G)$, we obtain a unique continuous extension $B(G) \to \bR$. This defines the integral of bounded functions.

Theorem: Every subgroup of an amenable group is amenable.

Proof: Let $G$ be amenable group with invariant mean $\mu$ and $H \leq G$. Choose a left transversal $L \subseteq G$ of $H$ and define

$$\nu\colon 2^H \to [0, 1], \quad A \mapsto \mu(LA).$$

Then $\nu$ has all the required properties. In particular $\nu(H) = \mu(LH) = \mu(G) = 1$.

Theorem: Every quotient of an amenable group is amenable.

Proof: Let $\mu$ be an invariant mean of $G$ and let $N \trianglelefteq G$ with projection $\pi\colon G \to G/N$. Then the pushforward measure

$$\nu\colon 2^{G/N} \to [0,1], \quad A \mapsto \mu(\pi^{-1}(A))$$

has the required properties.

Theorem: Let $1 \to N \to G \to G/N \to 1$ be an extension. Then $G$ is amenable if and only if both $N$ and $G/N$ are amenable.

Proof: Due to the previous results it suffices to show that if $N$ and $G/N$ are amenable, then so is $G$. Let $\mu$ and $\nu$ be invariants of $N$ and $G/N$ respectively. For each $A \subseteq G$ define

$$f_A\colon G \to [0, 1], \quad g \mapsto \mu(Ag^{-1} \cap N).$$

Notice that $f_A$ is constant on each coset of $N$. Indeed, if $x = ny$ for some $n \in N$ then

$$f_A(x) = \mu(Ax^{-1} \cap N) = \mu((Ay^{-1} \cap N)n^{-1}) = \mu(Ay^{-1} \cap N) = f_A(y).$$

Hence, the $f_A$ descend to the quotient $\bar f_A\colon G/N \to [0, 1]$. Since bounded functions are integrable, we can define an invariant mean for $G$ via

$$m\colon 2^G \to [0, 1], \quad A \mapsto \int_{G/N} \bar f_A \ d\nu.$$

This map has the desired properties due to the following facts.

1. $f_G = \mu(G (\cdot)^{-1} \cap N) = \mu(N) = 1$.
2. For disjoint $A,B \subseteq G$ we have $f_{A \cup B} = f_A + f_B$.
3. For $A \subseteq G$ and $g \in G$ we have $f_{Ag}(x) = f_A(xg^{-1})$.
4. $\nu$ is right-invariant and thus $\int_{G/N} \bar f_{A}(Nxg^{-1}) \ d\nu(Nx) = \int_{G/N} \bar f_{A}(Nx) \ d\nu(Nx)$

Corollary: $G$ and $H$ are amenable groups if and only if $G \times H$ is amenable.

Proof: Observe that $1 \to G \to G \times H \to H \to 1$ is an extension.

Theorem: Directed colimits of amenable groups are amenable.

Proof: Let $(G_i)_{i \in I}$ be a directed system of amenable groups with invariant means $(m_i)_{i \in I}$ and let $G = \colim G_i$. Using the cocone maps $f_i\colon G_i \to G$ we extend the $m_i$ to

$$\mu_i\colon 2^G \to [0, 1], \quad A \mapsto m_i(f_i^{-1}(A)).$$

To define the invariant mean of $G$ we use an appropriate ultrafilter limit. For $g \in G$ define

$$I_g = \{ i \in I : \mu_i(Ag) = \mu_i(A) \text{ for all } A \subseteq G \}.$$

The family $\{ I_g : g \in G \} \subseteq 2^I$ has the finite intersection property. Indeed, let $g_1, \dots, g_n \in G$. Each $g_k$ has a preimage under some $f_{i_k}\colon G_k \to G$. Since $I$ is directed, the set $\{ i_1, \dots, i_n \} \subseteq I$ has an upper bound $s \in I$ which satisfies $f_{i_k} = f_s \circ f_{i_k, s}$ so that $\mu_s$ is invariant under the right action of all $g_k$. We conclude $s \in \bigcap_{k=1}^n I_{g_k} \neq \varnothing$.

We can therefore extend the family $\{ I_g : g \in G \}$ to an ultrafilter $\mathcal U$ on $I$ and define

$$\mu\colon 2^G \to [0, 1], \quad A \mapsto \lim_{i \to \mathcal U} \mu_i(A).$$

The ultralimit inherits finite additivity and $\mu(G) = 1$ from the $\mu_i$ and since all $I_g \in \mathcal U$, we see that $\mu$ is right invariant under the action of $G$.

Corollary: All abelian groups are amenable.

Proof: Let $G$ be an abelian group. Then $G$ is the directed colimit of its finitely generated subgroups $(G_i)_{i \in I}$ ordered by inclusion. Each $G_i$ is of the form $\bZ^n \times \bZ/q_1\bZ \times \dots \times \bZ/q_r\bZ$. By the previous results $G_i$ is amenable if $\bZ$ is amenable. To construct the invariant mean on $\bZ$, let $F_n \coloneqq [-n, n] \cap \bZ$ and define

$$\mu_n \colon 2^{\bZ} \to [ 0, 1 ], \quad A \mapsto \frac{\abs{F_n \cap A}}{\abs{F_n}}.$$

It is easy to see that each $\mu_n$ is a finitely additive probability measure. Furthermore, we have

$$\begin{align*} \mu_n(A \pm 1) &= \frac{\abs{F_n \cap (A \pm 1)}}{\abs{F_n}} \leq \frac{\abs{(F_n \mp 1) \cap A}}{\abs{F_n}} \\ &\leq \frac{\abs{F_n \cap A}}{\abs{F_n}} + \frac{\abs{((F_n \mp 1) \setminus F_n) \cap A}}{\abs{F_n}} \\ &\leq \mu_n(A) + \frac{1}{2n+1} \end{align*}$$

We fix a non-principal ultrafilter $\mathcal U$ on $\bN$ and define

$$\mu\colon 2^\bZ \to [0, 1], \quad A \mapsto \lim_{n \to \mathcal U} \mu_n(A).$$

Then $\abs{\mu_n(A) - \mu_n(A \pm 1)} \to 0$ shows that $\mu$ is invariant under the group action.

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