How can I draw this Cayley graph?

Question

$G=\langle x,y,z:xy=yx,zy=yz,zx=xzy^2\rangle$. I want to draw the Cayley graph, we take as set of vertices $\mathbb{Z}^3\subset\mathbb{R}^3$, and make the correspondence $(i,j,k)\leftrightarrow x^iy^jz^k$. So by definition we have to connect $x^iy^jz^k$ with $x^iy^jz^kx,x^iy^jz^ky,x^iy^jz^kz$; now using the relations these words are equivalent respectively to $x^{i+1}y^{j+2k}z^k,x^iy^{j+1}z^k,x^iy^jz^{k+1}$. Now there is an easy way to draw this cayley graph? (if my calculation are correct).

Answer 1

The clearest way I can think of to "draw" this graph is to draw a layer of it by fixing a value of $k$. Take some "typical" value of $k$ like 3, draw it, and explain how it looks for general values of $k$. Then explain how the layers connect together (this looks straightforward; from your description of the group, the layers just connect vertically).