Regularity aspects for combinatorial simplicial surfaces

Baumeister, Markus; Niemeyer, Alice Catherine (Thesis advisor); Plesken gen. Wigger, Wilhelm (Thesis advisor)

Aachen (2020)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2020

Abstract

Combinatorial surfaces capture essential properties of continuous surfaces (like spheres and tori) in a discrete manner that lends itself more easily to a computational approach. They arise from triangulations of continuous surfaces and are based on an incidence structure between sets of vertices, edges, and faces. In this thesis, we focus on $\mathit{regularity}$ $\mathit{aspects}$. A combinatorial surface is regular if each vertex is incident to the same number of faces. Studying regular combinatorial surfaces is much easier than studying general combinatorial surfaces. A core idea of this thesis is transferring results for regular combinatorial surfaces to general combinatorial surfaces. The thesis contains four main projects: $\mathbf{(1)}$ A combinatorial surface $S$ can be represented by a $\mathit{net}$ of equilateral triangles in $\mathbb{R}^2$. This net can be interpreted as lying in a hexagonal lattice. The hexagonal lattice can be seen as a regular combinatorial surface $H$. Thus, a net can be described as a set of faces in $H$, together with pairs of edges in $H$, such that each of these pairs corresponds to one edge in $S$. We describe these pairs by automorphisms of $H$ and study the group generated by all these automorphisms. We show a correspondence between certain properties of $S$ (like orientability and existence of colourings) and properties of the generated group. $\mathbf{(2)}$ Modifications of combinatorial surfaces are often studied. We consider $\mathit{vertex}$ $\mathit{splits}$ of combinatorial surfaces with a single boundary and search for properties that are invariant under these modifications. To construct these properties, we extend the combinatorial surfaces along their boundary, such that every added vertex is incident to exactly six faces. This constructs the $\mathit{infinite}$ $\mathit{regular}$ $\mathit{extension}$, which remains unchanged under modifications of the original surface. Then, we classify combinatorial surfaces by the shapes of possible extensions. $\mathbf{(3)}$ A combinatorial surface can be constructed from a set of triangles, together with pairs of edges that should be identified (i. e. interpreted as the same edge). Each such pair allows a choice between two possible identifications. Changing all these choices simultaneously gives a different combinatorial surface, the $\mathit{geodesic}$ $\mathit{dual}$. We characterise those regular combinatorial surfaces that are isomorphic to their geodesic dual, by bringing them into correspondence with certain subgroups of triangle groups. In the finite cases (i. e. every vertex is incident to $d$ faces, with $d \leq 9$), we obtain a full classification. $\mathbf{(4)}$ The thesis is not purely theoretical. Together with Alice Niemeyer, software to support our research was developed: The $\mathtt{GAP}$-package $\mathtt{SimplicialSurfaces}$ encodes combinatorial surfaces and several common algorithms efficiently, allowing the user to focus on the underlying mathematical structure. Notable features include a library of surfaces that greatly facilitates testing of conjectures, and a flexible framework to build custom code for combinatorial surfaces, allowing for a wide variety of different research applications.