  We study the coset covering function $\mathfrak{C}(r)$ of a finitely
generated group: the number of cosets of infinite index subgroups needed to
cover the ball of radius $r$. We show that $\mathfrak{C}(r)$ is of order at
least $\sqrt{r}$ for all groups. Moreover, we show that $\mathfrak{C}(r)$ is
linear for a class of amenable groups including virtually nilpotent and
polycyclic groups, and that it is exponential for property (T) groups.
