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$G=\langle x,y,z:xy=yx,zy=yz,zx=xzy^2\rangle$. 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).

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It's kind of hard to "draw" a graph in $\mathbb Z^3$. – Patrick Da Silva Jan 29 '12 at 0:02
yeah I know but my homework ask me to draw it, so I thought that probably I was doing something wrong or there is an easy way to draw it. – John Jan 29 '12 at 0:33
I have never drawn Cayley graphs in my life, nor knew what was a Cayley graph before consulting Wikipedia : if my intuition is right from the stuff that I read, you should also connect $x^i y^j z^k$ with the dots obtained by multiplying on the right by $x^{-1}$, $y^{-1}$, $z^{-1}$, plus multiplication on the left... (and then see what cases you are left with.) Am I wrong, someone? – Patrick Da Silva Jan 29 '12 at 0:41
In any Cayley graph of a group with definite generators, it is suggested to identify elements of the group because; every generator wants to move them to the other element. It seems that this group is infinite, so it is hard to draw. Like this case, I think we should use van Kampen diagram instead. – Babak S. Jan 29 '12 at 10:46
Your correspondence $(i,j,k) \leftrightarrow x^i y^j z^k$ is unjustified. How do you know that every element of the group can be written in that form? – Lee Mosher May 22 '12 at 16:48
up vote 1 down vote accepted

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).

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