I realized that I know groups with distorted cyclic subgroups and groups all of whose free abelian subgroups are undistorted, but nothing between. Maybe it is a naive question, but:

Does there exist a group $G_n$ all of whose free abelian subgroups of rank $\leq n$ are undistorted and containing a distorted free abelian subgroup of rank $n+1$?

The case $n=2$ seems to be already interesting.

undistorted. Anyway, mapping tori of suitable automorphisms of $\mathbb{Z}^{n+1}$ should give examples for any $n$. $\endgroup$ – HJRW Dec 17 '14 at 16:25