Find the Cayley graph of the group $G = ( \mathbb{Z}/2 \mathbb{Z}) × ( \mathbb{Z}/2 \mathbb{Z})$ with generating set ${(1, 0),(0, 1)}$.

Question

I am quite new to geometric group theory and can't really visualize the way the Cayley graph changes when you change the generating set. I found many possible graphs here but I don't really see how to construct the one I am looking for.

Answer 1

Write out the four elements, clearly $(0,0)$ is connected to $(1,0),(0,1)$ and these are--in turn--each connected to $(0,0)$ and $(1,1)$ the first by adding themselves and the second by adding the other. And as each vertex has degree $2$ (Cayley graphs are regular graphs), that's all you need as $(1,1)$ already has both of it's connected vertices listed. So you end up with a square, the upper left corner being $(0,0)$, the two adjacent vertices $(1,0)$ and $(0,1)$, and the corner opposite $(0,0)$ is $(1,1)$.