Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 104 / 5 / c / Solution

Choose $1\ne h\in H$ and let $t$ generate $\mathbb{Z}$. For each binary string $({\epsilon }_0,\dots ,{\epsilon }_{n-1})$, the wordrecords that string in the lamps at positions $0,1,\dots ,n-1$. The resulting $2^n$ group elements are distinct and have word length at most $3n$ with respect to any finite generating set containing $h$ and $t$. Hence the growth function is bounded below exponentially, and $H\wr \mathbb{Z}$ has exponential growth.