Approximating a group by its solvable quotients

The solvable Farb growth of a group quantifies how well-approximated the group is 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 Farb 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 $Ck$th 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 Farb growth of some metabelian groups (including the Lamplighter groups).

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