A finitely generated subgroup $H$ of $G$ is said to be *un*distorted if any word metric on $H$ and any metric on $H$ induced by a word metric of $G$ are roughly equivalent (i.e., they differ by a multiplicative and additive constant; in other words, $H \hookrightarrow G$ is a quasi-isometric embedding). It is said to be distorted otherwise.

Baumslag–Solitar groups $G = \langle a, b \,|\, aba^{-1} = b^n \rangle$ provide easy example of groups $G$ with distorted, infinite cyclic, *non*-normal subgroups $H$. I was wondering how difficult it is to find examples of groups with distorted, infinite cyclic, *normal* subgroups.

distortedinfinite cyclic non-normal subgroups, right? $\endgroup$ – Anthony Quas Mar 30 '17 at 15:52