= Quadratic distortion of the Heisenberg center
{title2=$|c^m|_H\asymp\sqrt{|m|}$}
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 $m\geq0$, take $k=\lfloor\sqrt m\rfloor$ and write $m=k^2+r$, $0\leq r\leq2k$, to obtain length at most $4k+r\leq6\sqrt m$. 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 $m\ne0$. Negative exponents follow by inverses. Thus the ambient and intrinsic metrics have quadratic relative growth.
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