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.