I am interested in proving or disproving that certain Cayley graphs are expander.
Let $S$ be the multiplicative semigroup of matrices generated by $A = \left( \begin{array}{cc} a & b \\ 0 & 1 \end{array} \right)$ and $C = \left( \begin{array}{cc} c & d \\ 0 & 1 \end{array} \right)$ over $\mathbb{Z}/p\mathbb{Z}$, where $a,c\geq 2$, $A$ and $C$ don't commute, and $p$ is an odd prime.
Considering the Cayley graph $\Gamma_p(S, \{A,C\})$, is there a result that implies that $\Gamma_p$ is family of expander graphs? Any suggestions on how to prove it or disprove it?
Thanks in advance!
PS.: The semigroup generated by $A$ and $B$ is a free semigroup over $\mathbb{Z}$. So no product can ever be expressed more simply in terms of other elements, that is , there are no relations in the semgroup.