On the spectral properties of Cayley Graphs of metabelian and nilpotent group extensions

Let $(G_n)$ be a family of finite groups that are extensions of either metabelian groups or nilpotent groups of bounded class. In this paper, we show that $(G_n)$ cannot yield expander families of Cayley graphs. Denoting the Cayley graph of a group $G$ with respect to a set $S$ by $\Gamma(G,S)$, we further analyze the spectral properties of one such family that comprises the graphs $\mathcal{T}_{m,n,k} = \Gamma(\mathbb{Z}_m \ltimes_k \mathbb{Z}_n, \{(\pm 1,0),(0,\pm 1)\})$, where $k^m \equiv 1 \pmod{n}$. In addition to deriving some topological properties of this family, we show that if $m > 8$, then $\ST_{m,n,k}$ is not Ramanujan. Furthermore, we show that for a fixed $2 \leq m \leq 8$, $\mathcal{T}_{m,n,k}$ is not Ramanujan, for all $n \geq \lfloor e^{8cm^2} \rfloor$, where $c = 1/\log(2/\sqrt{3})$.