Quasi-isometrically embedded subgroup

A finitely generated subgroup $H\le G$ is quasi-isometrically embedded when its inclusion, equipped with word metrics from finite generating sets, is a quasi-isometric embedding. This property is independent of those generating sets.

A subgroup $H\le G$ is a retract when there is a group homomorphism $r:G\to H$ whose restriction to $H$ is the identity. If both groups are finitely generated, the inclusion of $H$ is a quasi-isometric embedding: the inclusion and retraction are both Lipschitz in their word metrics.

For a finitely generated subgroup $H\le G$, subgroup distortion compares the intrinsic word length $|h{|}_H$ with the ambient word length $|h{|}_G$. The subgroup is undistorted exactly when its inclusion is a quasi-isometric embedding.

This is the subgroup word-metric instance of distortion.

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.

A subgroup is exponentially distorted when elements with intrinsic length exponential in $n$ can have ambient length linear in $n$. In $\mathrm{BS}(1,2)$, the identity $b^{n}a{b}^{-n}=a^{2^n}$ makes the cyclic subgroup generated by $a$ exponentially distorted.