I've copied over this question from [what I asked on Mathematics Stack Exchange](https://math.stackexchange.com/q/4422568/976076), in the hope that some experts here can help me find a way to check the residual finiteness of this group.


Consider the group $G=K\rtimes \mathbb{Z}$ defined as follows:


The subgroup $K$ is generated by elements $x\_i,y\_k$ with $i,k \in {\mathbb Z}$ and $k > 0$, and it has defining relations
\begin{eqnarray\*}
 x\_i^2 &=& y\_j^2= 1\ \mbox{for all}\ i,j,\\
[x\_j,x\_i] &=& y\_{j-i}\ \mbox{for}\ j>i,\\ 
[y\_k,x\_i] &=& 1\ \mbox{for all}\ i,k,
\end{eqnarray\*}


The action of $({\mathbb Z},+)$ on $K$ is defined by the automorphism $1 \in {\mathbb Z}$ maps $x\_i$ to $x\_{i+1}$ for all $i \in {\mathbb Z}$.


**Question**: Is group $G$ residually finite?


**The progress**: My idea is to check if $K$ is residually finite first (because if $K$ is not residually finite, then $G$ can't be). So far, if a word $w$ from $K$ satisfies the following condition, then there is a homomorphism from $K$ to a finite group that doesn't send $w$ to the identity.


* if there exists $x\_i$ in $w$, and the total power of $x\_i$ is odd. (we can map $K$ to some direct product of $\mathbb{Z}\_2$)
* if $w= y\_j$ and $j$ is odd. (We can map $K$ to the Heisenberg group over $\mathbb{Z}\_2$)


I am not sure how to show such homomorphism exists for any general word. (e.g. a string of $y\_i$'s).



Is this semi-direct product residually finite?
Yes, it's residually finite.


This group maps onto the residually finite wreath product $C\_2\wr\mathbf{Z}$ and the kernel is central, free over the $y\_k$, $k\ge 0$, as 2-elementary abelian group.


So one needs to show that for every non-empty subset $J$ of the set $J$ of positive integers, the element $y\_J=\prod\_{j\in J}y\_j$ survives in some finite quotient.


Let $t$ be the generator of $\mathbf{Z}$ and kill $t^n$. This kills $y\_n$ and identifies $y\_i$ to $y\_{i+n}$ for all $n$ (and identifies $x\_i$ to $x\_{i+n}$. At the level of the quotient, the central kernel of the homomorphism onto $C\_2\wr C\_n$ is freely generated by the $y\_i$, $0<i<n$, as $2$-elementary abelian group. In particular, $y\_J$ is not trivial in this quotient.


