Trivalent expanders, $(\Delta –Y)$-transformation, and hyperbolic surfaces

Abstract

We construct a new family of trivalent expanders tessellating hyperbolic surfaces with large isometry groups. These graphs are obtained from a family of Cayley graphs of nilpotent groups via $(Delta–Y)$-transformations. We study combinatorial, topological and spectral properties of our trivalent graphs and their associated hyperbolic surfaces. We compare this family with Platonic graphs and their associated hyperbolic surfaces and see that they are generally very different with only one hyperbolic surface in the intersection. Finally, we provide a number theory free proof of the Ramanujan property for Platonic graphs and a special family of subgraphs.