Exponential growth of the ternary affine Baumslag-Solitar group ($⟨x,y\mid x^{-1}yx=y^3⟩$)

Represent $y$ by translation $t\mapsto t+1$ and $x$ by $t\mapsto t/3$; the relator holds and $y$ has infinite order. Base-three digits $d_j\in \{0,1,2\}$ give words recursively by $W_0=1$, $W_{j+1}=x^{-1}{W}_{j}x{y}^{d_j}$. These have length at most $4k$ and represent all $y^m$, $0\le m<3^k$. Thus the growth function of a finitely generated group obeys $\beta (4k)\ge 3^k$. Counting freely reduced words in four letters gives $\beta (n)\le 2\cdot 3^n-1$. The growth is therefore exponential up to coarse equivalence.