Geometry & Topology: Reymond Akpanya -- On face-transitive simplicial surfaces

Monday 4 September 2023, 16:00-17:00, Carslaw 535

Please note that this is a second G&T seminar on this Monday. We farewell Reymond, 
who visited Jonathan Spreer for an extended period!

Speaker: Reymond Akpanya (RWTH Aachen)

Title: On face-transitive simplicial surfaces

Abstract: A simplicial surface can be derived from the incidence geometry of a 
triangulated three-dimensional polyhedron by denoting the relationships between 
the corresponding vertices, edges and faces. We refer to a simplicial surface as 
face-transitive if the corresponding automorphism group acts transitively on the 
faces of the simplicial surface.

We link a simplicial surface to its dual graph, a cubic graph, by denoting the 
incidences between its faces and edges. While translating a surface into a cubic 
graph is straightforward, determining whether there exists a simplicial surface with 
a given cubic graph as dual graph is a task of high complexity. However, when dealing 
with vertex-transitive cubic graphs, there is some optimism in tackling this challenge.
In this talk, we will discuss the computation of face-transitive surfaces that have 
vertex-transitive cubic graphs as their dual graphs. We will demonstrate that, in the 
case of a vertex-transitive cubic graph, the necessary information for this construction 
can be recovered from the automorphism group of the cubic graph. Moreover, we will see 
that there are exactly 11 different types of face-transitive surfaces that can occur and 
also provide distinguishing invariants to tell them apart.