Explore a 52-minute lecture on the coset covering function of infinite, finitely generated groups. Delve into the study of C(r), which represents the number of cosets of infinite index subgroups needed to cover a ball of radius r. Discover the proof that C(r) is of order at least √r for all groups. Examine the linear behavior of C(r) for a class of amenable groups, including virtually nilpotent and polycyclic groups, and contrast this with its exponential nature for property (T) groups. This talk, part of the Measured Group Theory program at the Centre de recherches mathématiques, offers insights into a quantitative version of Neumann's lemma for finitely generated groups.