Cayley graph of an amalgam

Question

I was wondering what the Cayley graph of an amalgam $A*B$ (over $\{ 1\}$ ) is?

$A, B$ are finitely generated. Can't work it out.

Answer 1

Let $\Gamma_A$ and $\Gamma_B$ be the Cayley graphs of $A$ and $B$, respectively. The Cayley graph of $A*B$ is constructed by an infinite iterative process where many copies of $\Gamma_A$, $\Gamma_B$ are glued together. Start with one copy of $\Gamma_A$. At each of vertex $v$ of $\Gamma_A$, glue $v$ to one vertex of a new copy of $\Gamma_B$. At every remaining vertex of each of these new copies of $\Gamma_B$, glue that vertex to one vertex of a new copy of $\Gamma_A$. At every remaining vertex of each of these new copies of $\Gamma_A$, glue that vertex to one vertex of a new copy of $\Gamma_B$. And so on, and so on …