h1

% IMPORTANT: The following is UTF-8 encoded.  This means that in the presence
% of non-ASCII characters, it will not work with BibTeX 0.99 or older.
% Instead, you should use an up-to-date BibTeX implementation like “bibtex8” or
% “biber”.

@PHDTHESIS{Baumeister:793575,
      author       = {Baumeister, Markus},
      othercontributors = {Niemeyer, Alice Catherine and Plesken gen. Wigger, Wilhelm},
      title        = {{R}egularity aspects for combinatorial simplicial surfaces},
      school       = {RWTH Aachen University},
      type         = {Dissertation},
      address      = {Aachen},
      reportid     = {RWTH-2020-07027},
      pages        = {1 Online-Ressource (235 Seiten) : Illustrationen},
      year         = {2020},
      note         = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
                      University; 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.},
      cin          = {115320 / 110000},
      ddc          = {510},
      cid          = {$I:(DE-82)115320_20140620$ / $I:(DE-82)110000_20140620$},
      typ          = {PUB:(DE-HGF)11},
      doi          = {10.18154/RWTH-2020-07027},
      url          = {https://publications.rwth-aachen.de/record/793575},
}