The study of groups often sheds light on problems in various areas of mathematics. Whether playing the role of certain invariants in topology, or encoding symmetries in geometry, groups help us understand many mathematical objects in greater depth. In coarse geometry, one can use groups to construct examples or counterexamples with interesting or surprising properties. In this talk, we will introduce one such metric object arising from finite quotients of finitely generated groups, and survey some of its useful properties and associated constructions.[-]