= Exponential growth of the ternary affine Baumslag-Solitar group
{title2=$\langle x,y\mid x^{-1}yx=y^3\rangle$}
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 $W_0=1$, $W_{j+1}=x^{-1}W_jxy^{d_j}$. 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 $\beta(4k)\ge3^k$. Counting freely reduced words in four letters gives $\beta(n)\le2\cdot3^n-1$. The growth is therefore exponential up to coarse equivalence.
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