= Exponentially distorted subgroup
For <finitely generated groups> $H\leq G$, choose finite symmetric <generating sets> $T$ for $H$ and $S$ for $G$, and measure <subgroup distortion> by
$$
\Delta_H^G(n)=\max\{|h|_T:h\in H,\ |h|_S\leq n\}.
$$
The 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 group>
$$
G=\langle a,t\mid tat^{-1}=a^2\rangle,
$$
the <cyclic subgroup> $H=\langle a\rangle$ is exponentially distorted. The identity $t^kat^{-k}=a^{2^k}$ 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 $|m|\leq n2^n$. These matching bounds establish the exponential distortion class, independently of the chosen finite <generating sets>.
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