Let $G$ be a finitely presented group with polynomial growth. Fix a generating set for $G$.
For every subgroup $H$ of $G$, element $g\in G$ and positive number $R$, define $\alpha(H,g,R)$ to be $|\partial B(gH,R)|/|B(gH,R)|$ where $B(gH,R)$ is the ball in the Schreier graph $G/H$ with center $gH$ and radius $R$.
For every positive number $R$, define $\alpha(R)$ as the supremum of the $\alpha(H,g,R)$ where $H$ runs over all finite index subgroups of $G$ and $g$ runs over all elements of $G$.
Is there a sequence for radii $R_1,R_2,R_3,\dotsc$ tending to infinity such that $\alpha(R_i)$ tends to zero as $i$ tends to infinity?