Which is the Cayley Graph of $\mathbb{Z}_2 * \mathbb{Z}_2$

Question

I am trying to figure out how is the Cayley graph of the group $\mathbb{Z}_2 * \mathbb{Z}_2$.

I think it should be a infinite tree, but I'm not sure because if $a$ and $b$ are the generators of this group and we consider the product $a*a$ (does it make sense to consider this product?) then $$ a * a = e, $$ so, should be and edge from $a$ to $e$? There is already an edge from $e$ to $a$, so in that case, we'll have a cycle in the graph...

(by the product $*$, I mean yuxtaposition with cancellation)

P.S.: I'm interested if defining and order would be valid, like in te free group of $2$ generators $\mathbb{F}_2$, in which, for example $a^2 b a \leq a^2 b a b$. In the group $\mathbb{Z}_2 * \mathbb{Z}_2$, $e \leq a$ or $a \leq e$??

Answer 1

This group is just the infinite diedral group action on a line ($\bf R$). Let see why the Cayley graph is just this line, with vertices the point $n+{1\over 2}, n\in \bf Z$ and edges the segemnts $[n+{1\over 2}, n+{1\over 2}]$. Indeed, let $a$ be the symmetry $x\to -x$, and $b$ the symmetry $x\to -x+2$. Then $a,b$ are of order 2, preserve this graph. We have two orbits of edges (the edges $[n+{1\over 2}, n+{1\over 2}]$, $n $ even, and $n$ odd (note that $a=a^{-1}, n=b^{-1}$). So this graph is marked $abababab..$., So the infinite diedral group operate on this graph, exactly like in its Cayley graph.

Answer 2

As usual, I find that sitting down and drawing a picture can be helpful. Notice that words in $\mathbb{Z}_2\ast\mathbb{Z}_2$ only come in one form: alternating between single powers of $a$ and $b$. For example, there are only two words of length six: $ababab$ and $bababa$.

So, let’s draw. I’ll have the $a$-edges be solid lines, and $b$-edges be dotted lines:

You can see that this will extend to the infinite $2$-regular tree.