Quantifying residual finiteness

We introduce the notion of quantifying the extent to which a finitely generated group is residually finite. We investigate this behavior for examples that include free groups, the first Grigorchuk group, finitely generated nilpotent groups, and certain arithmetic groups such as SLn(Z). In the context of finite nilpotent quotients, we find a new characterization of finitely generated nilpotent groups.