A quantitative Neumann lemma for finitely generated groups

TL;DR

This paper investigates the growth of the coset covering function in finitely generated groups, establishing lower bounds and classifying its behavior for different group classes, revealing new insights into group coverings.

Contribution

It introduces a quantitative version of Neumann's lemma, providing bounds on the coset covering function for various classes of finitely generated groups.

Findings

(r)    r^{1/2} for all groups

(r) is linear for amenable groups including virtually nilpotent and polycyclic groups

(r) is exponential for property (T) groups

Abstract

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.