\infty} \left(
\frac{ \sharp(T[q_m(F)]) }{ \sharp(q_m(F)) } \right) < 1 + \epsilon.\end{aligned}$$ There exists a controlled set $\widetilde{T}$ such that for every $m$ the finite subset $q_m(F)$ is $\widetilde{T}$-bounded. Indeed, there exists a natural number $l$ such that $F \subseteq K^l$. By the inclusion $q_m(F) \subseteq T_K^{\circ l} [1_{G/H_m}]$, the subsets $q_m(F)$ are $T_K^{\circ l}$-bounded. It follows that the box space $\square G$ is not a sequence of weak expander spaces.

Conversely, suppose that $\square G$ is not a sequence of weak expander spaces. For every positive number $\epsilon$, there exists a controlled set $\widetilde{T}$ satisfying the following inequality: $$\begin{aligned}
\liminf_{m \rightarrow \infty} \left(
\inf \left\{ \frac{ \sharp(T_K[Y]) }{ \sharp(Y) } \ \colon\ 
\emptyset \neq Y \subseteq G / H_m, Y \textrm{\ is\ a } 
\widetilde{T} \textrm{-bounded\ set} \right\} \right) 
< 1 + \epsilon.\end{aligned}$$ We may assume that $\widetilde{T} = T_K^{\circ n}$. For large enough $m$, the quotient map $q_m \colon G \rightarrow G / H_m$ is injective on $K^{n + 1} \subseteq G$. Take $m$ such that $q_m |_{K^{n + 1}}$ is injective and that $$\begin{aligned}
\inf \left\{ \frac{ \sharp(T_K[Y]) }{ \sharp(Y) } \ \colon\ 
\emptyset \neq Y \subseteq G / H_m, Y \textrm{\ is\ a } 
T_K^{\circ n} 