A quantitative Neumann lemma for finitely generated groups

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.