Suppose $F_2 = \mathbb{Z} * \mathbb{Z} $ and $ \Gamma_S$ be the Cayley Graph of $F_2$ over the usual generating set.
So we have a usual 4-regular tree as $ \Gamma_S $
Suppose $ \mathbb{Z} $ act on $ \Gamma_S $ by left action. How does the quotient graph $ \mathbb{Z} / \Gamma_S $ look like?