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Problem (An Infinite Cayley Graph)
In each scenario below, you should draw a Cayley graph.
- Let $X=\mathbb{Z}$, an infinite set. Consider the permutation $\phi_1:\mathbb{Z}\to \mathbb{Z}$ defined by $\phi_1(x)=x+1$. This permutation shifts elements in $\mathbb{Z}$ right one, with no wrap around. If we let $S=\{\phi_1\}$, give a rough sketch of the Cayley graph of $\text{span}(S)$. It's a rough sketch of the graph because there are infinitely many vertices.
- How would you change your graph in part 1 if instead you used $S=\{\phi_2,\phi_3\}$, where these are shifts right 2 and right 3 respectively?
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