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@PHDTHESIS{Akpanya:1016413,
author = {Akpanya, Reymond Oluwaseun},
othercontributors = {Niemeyer, Alice Catherine and Spreer, Jonathan and Robertz,
Daniel},
title = {{C}onstructing symmetric simplicial surfaces},
school = {RWTH Aachen University},
type = {Dissertation},
address = {Aachen},
publisher = {RWTH Aachen University},
reportid = {RWTH-2025-06914},
pages = {1 Online-Ressource : Illustrationen},
year = {2025},
note = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
University; Dissertation, RWTH Aachen University, 2025},
abstract = {This thesis studies triangulated 2-dimensional manifolds
from a combinatorial perspective. We explore the properties
of such triangulations with the help of simplicial surfaces.
Here, a simplicial surface encodes the incidence relations
between the vertices, edges and triangles of a triangulated
2-dimensional manifold, e.g., a disc, a sphere or a torus.
For instance, some of the Platonic solids, namely the
tetrahedron, the octahedron and the icosahedron, can be
understood as triangulations of the 2-sphere and hence can
be translated into simplicial surfaces. Since each triangle
(face) of a simplicial surface is incident to three edges,
these combinatorial structures can be linked to cubic
graphs. While constructing a cubic graph from a given
simplicial surface is computationally easy, deciding whether
a given cubic graph is the graph describing the incidences
between the faces and edges of some simplicial surface is
quite challenging; even more, the existence of such a
simplicial surface for an arbitrary cubic graph is still an
open problem in graph-theoretical research.The goal of this
thesis is to study simplicial surfaces and their connections
to cubic graphs. Moreover, we provide construction methods
for cubic graphs and simplicial surfaces arising from cubic
graphs. To achieve this goal, we make use of
group-theoretical approaches. First, we examine
homomorphisms between simplicial surfaces. Of particular
interest are simplicial surfaces that have no proper
epimorphic images. That means we study simplicial surfaces,
where every epimorphism from such a simplicial surface onto
another simplicial surface is already an isomorphism. We
establish the existence of infinitely many simplicial
surfaces that satisfy the above property and further
elaborate on special homomorphisms, namely endomorphisms and
butterfly-friendly homomorphisms between simplicial
surfaces.As another main result, we modify a well-known
cubic graph construction, presented by Frucht in 1949, and
exploit our modification to establish the existence of
simplicial surfaces having automorphism groups that are
isomorphic to arbitrary finite groups. For certain groups,
we further associate the arising simplicial surfaces to
polyhedra in Euclidean 3-space such that the simplicial
surfaces and the corresponding polyhedra have isomorphic
automorphism and symmetry groups. Next, we exploit highly
symmetric cubic graphs to construct simplicial surfaces with
large automorphism groups. More precisely, we demonstrate
that the automorphism group of a vertex- or edge-transitive
cubic graph can be used to construct symmetric simplicial
surfaces such as closed face-transitive or closed
edge-transitive surfaces. These constructions are achieved
by providing suitable cycle double covers of given vertex-
or edge-transitive cubic graphs. Our theoretical results
allow us to compile two censuses: one containing closed
face-transitive surfaces with at most 1280 faces, derived
from cubic vertex-transitive graphs with automorphism groups
of order at most $10^11,$ and one containing closed
edge-transitive surfaces with at most 5000 faces,
constructed from cubic edge-transitive graphs. Finally, we
classify the sets of open face-transitive surfaces and open
edge-transitive surfaces.},
cin = {115320 / 110000},
ddc = {510},
cid = {$I:(DE-82)115320_20140620$ / $I:(DE-82)110000_20140620$},
pnm = {DFG project G:(GEPRIS)444414437 - Algebra, Kinematik und
Kompatibilität triangulierter Geometrien (A04) (444414437)
/ DFG project G:(GEPRIS)417002380 - TRR 280:
Konstruktionsstrategien für materialminimierte
Carbonbetonstrukturen – Grundlagen für eine neue Art zu
bauen (417002380)},
pid = {G:(GEPRIS)444414437 / G:(GEPRIS)417002380},
typ = {PUB:(DE-HGF)11},
doi = {10.18154/RWTH-2025-06914},
url = {https://publications.rwth-aachen.de/record/1016413},
}
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