A quantitative Neumann lemma for finitely generated groups

Elia Gorokhovsky, Nicolás Matte Bon, Omer Tamuz

Published 2022-03-21Version 1

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 linear for virtually nilpotent groups, exponential for property (T) groups, and is of order at least $\sqrt{r}$ for all groups.