= Dyadic affine Baumslag-Solitar group
{title2=$\operatorname{BS}(1,2)\cong\mathbb Z[1/2]\rtimes\mathbb Z$}
In the <finite group presentation> $\langle a,t\mid tat^{-1}=a^2\rangle$, let $a$ act by $x\mapsto x+1$ and $t$ by $x\mapsto2x$. The conjugates $a_i=t^{-i}at^i$ satisfy $a_{i+1}^2=a_i$, so their increasing cyclic subgroups give the additive group $\mathbb Z[1/2]$. Every element has the form $a_i^m t^k$, and the affine action proves that this form can be the identity only if $k=m=0$. Thus the displayed <semidirect product> identification holds. This is a finitely presented group with a series having torsion-free abelian factors, but it is not a <poly-infinite cyclic group>, since $\mathbb Z[1/2]$ is not finitely generated.
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