The solvable residual finiteness growth of a group quantifies how well the group is approximated by its finite solvable quotients. In this note we present a new characterization of polycyclic groups which are virtually nilpotent. That is, we show that a group has solvable residual finiteness growth which is at most polynomial in log(n) if and only if the group is polycyclic and virtually nilpotent. We also give new results concerning approximating oriented surface groups by nilpotent quotients. As a consequence of this, we prove that a natural number C exists so that any nontrivial element of the Ckth term of the lower central series of a finitely generated oriented surface group must have word length at least k. Here C depends only on the choice of generating set. Finally, we give some results giving new lower bounds for the solvable residual finiteness growth of some metabelian groups (including the Lamplighter groups).

The author was partially supported by NSF RTG grant DMS-0602191.