Computational Algebra Seminar: Akpanya -- On the Construction of Edge-transitive Surfaces

Speaker: Reymond Akpanya (Magma)
Title: On the Construction of Edge-transitive Surfaces
Time & Place: 15.00-16.00, Thursday 13 November, SMRI Seminar Room
Abstract: A simplicial surface can be seen as the incidence geometry of the vertices,
edges and faces of a triangulated 2-manifold. We call such a surface edge-transitive
if its automorphism group acts transitively on the edges of the surface. A given 
simplicial surface can be linked to a cubic graph by recording the incidences between 
the corresponding faces and edges. The resulting cubic graph does not directly contain 
any information on the vertices of the corresponding surface. This missing information 
is obtained by constructing a cycle double cover of the corresponding cubic graph, 
i.e. a collection of cycles such that every edge of the graph lies in exactly two cycles.

In this talk, we discuss the construction of edge-transitive surfaces by providing 
suitable cycle double covers of edge-transitive cubic graphs. We show that there exist 
four types of edge-transitive surfaces, splitting up further into a total of five 
sub-types.  We exploit our theoretical results to compute a census of edge-transitive 
surfaces with up to 5000 faces.