Algebra Seminar: Goertzen -- Simplicial Surfaces with given Automorphism Group

Tom Goertzen (University of Sydney) will be speaking in the algebra seminar.  We will go
out for lunch after the talk---all are welcome to join! 

When: 12-1pm Friday August 15 

Where: SMRI Seminar Room (Macleay Building A12 Room 301) 

Title: Simplicial Surfaces with given Automorphism Group 

Abstract: 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.