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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.

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.

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.