Abstract:

Frucht demonstrated that every finite group can be realised as the automorphism group of a cubic graph (every node has three edges). In this talk, we refine Frucht’s construction for groups generated by two elements, producing cubic graphs with fewer vertices. For arbitrary finite groups, we identify and rectify an oversight in the original construction. We establish the existence of cycle double covers in these cubic graphs by utilising edge colourings, which lead to simplicial surfaces with given finite automorphism groups. This is achieved through a correspondence between simplicial surfaces and cubic graphs equipped with cycle double covers. Additionally, we derive alternative cycle double covers that offer a discrete perspective on the classical Riemann–Hurwitz formula. If time permits, we will explore alternative constructions of surfaces with smaller face counts and investigate embeddings of certain surfaces into Euclidean three-space where every edge has unit length. This is joint work with Reymond Akpanya.