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.

For finitely generated groups $H\le G$, choose finite symmetric generating sets $T$ for $H$ and $S$ for $G$, and measure subgroup distortion byThe subgroup is exponentially distorted when this function has exponential growth up to the usual coarse comparison. In particular it has both an exponential lower bound and an exponential upper bound after rescaling the argument; an exponential lower bound alone could also occur for more severely distorted subgroups.

For the dyadic affine Baumslag-Solitar groupthe cyclic subgroup $H=⟨a⟩$ is exponentially distorted. The identity $t^{k}a{t}^{-k}=a^{2^k}$ gives ambient word length at most $2k+1$ and intrinsic word length $2^k$.

For the upper bound, use its faithful group action on the real line, given by $a(x)=x+1$, $t(x)=2x$. A word of length at most $n$ representing $a^m$ has at most $n$ translation contributions, each of absolute size at most $2^n$. Thus $|m|\le n{2}^n$. These matching bounds establish the exponential distortion class, independently of the chosen finite generating sets.