  In this paper, we mainly elucidate a close relationship between the
topological entropy and mean dimension theory for actions of polynomial growth
groups. We show that metric mean dimension and mean Hausdorff dimension of
subshifts with respect to the lower rank subgroup are equal to its topological
entropy multiplied by the growth rate of the subgroup. Meanwhile, we also prove
the above result holds for the rate distortion dimension of subshifts with
respect to the lower rank subgroup and measure entropy. Furthermore, some
relevant examples are indicated.
