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Distortion measures how a mapping changes distances or geometric quantities relative to their original values. For an injective map between metric spaces, one multiplicative measure is the product of its Lipschitz constant and the Lipschitz constant of its inverse on the image. Subgroup distortion instead compares intrinsic and ambient word metrics through a growth function.

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.