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.



Are there known and interesting or easy examples of groups $G$ which admit $\mathbb Z$ as a distorted normal subgroup?
As explained by A. Sisto [here](https://people.math.ethz.ch/~alsisto/LectureNotesGGT.pdf) on p.20 and p.21, "The subgroup generated by
$z$
in the Heisenberg group
$$〈
x,y,z
|
[
x,y
] =
z,
[
x,z
] = [
y,z
] = 1
〉$$
is isomorphic to
$\mathbf Z$
and distorted." As $\langle z \rangle$ is the center of the Heisenberg group and hence normal, this gives one example. 


