\textrm{-bounded\ set} \right\} < 1 + \epsilon.\end{aligned}$$ Choose a $T_K^{\circ n}$-bounded subset $Y \subseteq G / H_m$ satisfying $\sharp(T_K[Y]) < (1 + \epsilon) \sharp(Y)$. Replacing $Y$ with a right translation of $Y$, we may assume that $1_{G/H_m} \in Y$. Since $T_K[Y]$ is $T_K^{\circ (n+1)}$-bounded, $T_K[Y]$ is included in the image of $K^{n + 1}$. The inverse image $F$ of $Y$ under the injective map $q_m |_{K^{n + 1}}$ satisfies the inequality $$\begin{aligned}
\sharp(K F) < (1 + \epsilon) \sharp(F).\end{aligned}$$ It turns out that there exists an $\epsilon$-Følner set with respect to the generating set $K$. It follows that $G$ is amenable.

Expander and weak expander {#SubsectionExpander}
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Let $\{(X_m, d)\}_{m \in \mathbb{N}}$ be a sequence of finite metric spaces. The disjoint union $\bigsqcup_{m = 1}^\infty X_m$ is equipped with a coarse structure defined by $$\begin{aligned}
\mathcal{C} =
\left\{\left.
T \subseteq \bigsqcup X_m \times X_m \ \right| \ 
d \mathrm{\ is\ bounded\ on\ } T \right\}.\end{aligned}$$ Suppose that $\left( \bigsqcup_{m = 1}^\infty X_m, \mathcal{C} \right)$ is a uniformly locally finite coarse space. Definition \[DefinitionWeakExpander\] can be rephrased in terms of the metric $d$. The space $\bigsqcup_{m = 1}^\infty (X_m, d)$ is a sequence of weak expander spaces if and only if 