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

  • Ioannis Ivrissimtzis

    Durham University, UK
  • Norbert Peyerimhoff

    Durham University, UK
  • Alina Vdovina

    University of Newcastle, Newcastle upon Tyne, UK
Trivalent expanders, $(\Delta – Y)$-transformation, and hyperbolic surfaces cover
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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 (DeltaY)(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.

Cite this article

Ioannis Ivrissimtzis, Norbert Peyerimhoff, Alina Vdovina, Trivalent expanders, (ΔY)(\Delta – Y)-transformation, and hyperbolic surfaces. Groups Geom. Dyn. 13 (2019), no. 3, pp. 1103–1131

DOI 10.4171/GGD/518