= Iterated endomorphism kernels in BS(2,3)
{title2=$G''\subseteq\bigcup_{k\ge1}\ker\theta^k\subseteq R_G$}
For $G=\langle a,t\mid ta^2t^{-1}=a^3\rangle$, the surjective endomorphism $\theta(a)=a^2$, $\theta(t)=t$ has nontrivial kernel by <Britton's lemma>. The <commutator subgroup> is generated by $a_i=t^iat^{-i}$ for $i\in\mathbb Z$. For $j>i$, its defining relations give $a_j^{2^{j-i}}=a_i^{3^{j-i}}$. Hence $\theta^k([a_i,a_j])=1$ whenever $k\ge j-i$. The union of the increasing kernels is normal, and the images of all $a_i$ commute in its quotient, proving the first inclusion. The second follows from <finite residual contains surjective endomorphism kernels> applied to every iterate.
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