Full residual finiteness growths of nilpotent groups

Full residual finiteness growth of a finitely generated group G measures how efficiently word metric n-balls of G inject into finite quotients of G. We initiate a study of this growth over the class of nilpotent groups. When the last term of the lower central series of G has finite index in the center of G we show that the growth is precisely nb, where b is the product of the nilpotency class and dimension of G. In the general case, we give a method for finding an upper bound of the form nb where b is a natural number determined by what we call a terraced filtration of G. Finally, we characterize nilpotent groups for which the word growth and full residual finiteness growth coincide.