University of Sydney Algebra Seminar

Tom Goertzen

Friday 15 August, 12-1pm, in SMRI Seminar Room (Macleay Building A12 Room 301)

Simplicial Surfaces with given Automorphism Group

In this talk, we will establish the existence of simplicial surfaces with given finite automorphism group. Simplicial surfaces can be defined as the incidence structure of triangulated surfaces. By defining a cubic graph in which the vertices correspond to the faces of such a surface, the underlying incidence structure corresponds to a cycle double cover of this graph, i.e. a collection of cycles such that each edge is contained in exactly two cycles. Frucht demonstrated that every finite group can be realised as the automorphism group of a cubic graph. 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. Additionally, we derive alternative cycle double covers that offer a discrete perspective on Hurwitz's automorphisms theorem. 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.