A quantitative Neumann lemma for finitely generated groups

We study the coset covering function ℭ(r) of an infinite, finitely generated group: the number of cosets of infinite index subgroups needed to cover the ball of radius r. We show that ℭ(r) is of order at least \(\sqrt{r}\) $$\sqrt{r}$$ for all groups. Moreover, we show that 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.