Let $\{ G_n \}_{n \ge 1}$ be a sequence of graphs such that the number of vertices of $G_n$ tends to $\infty$ as $n \to \infty$. We say that $\{ G_n \}_{n \ge 1}$ is an expander family if $\lambda_2( G_n)$ is bounded away from $0$ as $n \to \infty$. Here $\lambda_2(H)$ is the smallest positive eigenvalue of the normalized graph Laplacian of $H$.

If every $G_n$ is the $1$-skeleton of a simple $3$-polytope, you can not have an expander family. This follows from the planar separator theorem.

What if every $G_n$ is the $1$-skeleton of a simple $4$-polytope? If you relax "simple" then there are obvious examples — in particular, cyclic polytopes in dimension $4$ have complete graphs for their $1$-skeletons, so they are expanders.

If it is hard to come up with an example for simple $4$-polytope, does it make it easier if we allow the dual graph of an arbitrary simplicial $3$-sphere?