 Exponentially distorted subgroup

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.