Quadratic distortion of the Heisenberg center ($|c^m{|}_H\asymp \sqrt{|m|}$)

The central infinite cyclic subgroup of the Integer Heisenberg group has quadratic subgroup distortion. The identity $c^{k^2}=[a^k,b^k]$ gives ambient length at most $4k$ against intrinsic length $k^2$. For general $m\ge 0$, take $k=\lfloor \sqrt{m}\rfloor$ and write $m=k^2+r$, $0\le r\le 2k$, to obtain length at most $4k+r\le 6\sqrt{m}$. Conversely, a word of length $n$ in $a,b,c$ changes the first coordinate by at most $n$ and its central coordinate by at most $n^2+n$, so $|c^m|$ is bounded below by a positive multiple of $\sqrt{|m|}$ for $m\ne 0$. Negative exponents follow by inverses. Thus the ambient and intrinsic metrics have quadratic relative growth.