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## Research## Visualizing Regular Tessellations: Principal Congruence Links and Equivariant Morphisms from Surfaces to 3-Manifolds (PhD Thesis, PDF, Errata)The PhD Thesis has two parts:## Principial Congruence Links (Slides of Talk)Thurston gave an example of an 8-component link whose complement is covered by 24 regular ideal hyperbolic tetrahedra such that the symmetries of the link complement can take any tetrahedra to any other tetrahedra in all possible 12 orientations of the tetrahedra. I show how to construct two more links with 12 respectively 20 components with this special property. These links are examples of prinicipal congruence links, i.e., links whose complement is H ^{3} divided by a principal congruence subgroup of PGL(2,O_{D}) where O_{D} is the ring of integers of the imaginary quadratic number field of discrimninant D=-3. The groups PGL(2,O_{D}), respectively, PSL(2,O_{D}) (Bianchi groups) are of special interest because every arithmetic cusped hyperbolic 3-manifold is commensurable with such a group for some D. The principal congruence subgroup are the easiest to define subgroups of PGL(2,O_{D}). They are given by ker(PGL(2,O_{D}) ->PGL(2,O_{D}/I) for some ideal I, yielding an interesting class of arithmetic cusped hyperbolic 3-manifolds with very large symmetry group.
## Equivariant Morphisms Regular Maps to 3-Manifolds
The second chapter gives an algorithm to determine how much symmetry of a surface F can be seen by mapping, immersing, or embedding F into Euclidean 3-space E ^{3}. Here, F is a
"regular map" as dened by, e.g., Coxeter and is the generalization of the Platonic
solids to higher genus surfaces, the genus 0 regular maps being exactly the surfaces of the
Platonic solids. In this denition, the term "map" refers to a tessellation of a surface (as
in countries of a geographic map) and a "regular map" is a tessellation by p-gons such that
q of them meet at each vertex and fulll an extra transitivity condition. Notice that any
automorphism of the surface of a Platonic solid regarded as regular map is also realized by
an isometry of the Platonic solid regarded as solid in E^{3}. However, embeddings of most
higher-genus regular maps fail to make all symmetries directly visible in space. The Klein
quartic is a regular map of genus 3 tessellated by heptagons and an embedding of it into E^{3}
is visualized by the sculpture "The Eightfold Way". We cannot rotate the sculpture so that
one heptagon is rotated by a 1/7th of a turn, even though this rotation is induced abstractly
by an automorphism of the Klein quartic. However, the symmetries of the tetrahedron
form a subgroup H of automorphisms that are visible in the sculpture. This gives rise to
the question what the best sculpture for a given regular map F is in terms of symmetries
directly made visible in space. We present algorithms to determine which subgroups H of the
automorphism group of a given regular map F are realized by an H-equivariant morphism,
immersion, or embedding into E^{3}. We show the results for the census of regular maps by
Conder and Dobcsanyi up to genus 101.To achieve this, we translate the question about the existence of an equivariant morphism into the existence of morphisms between the quotient spaces of F and E ^{3} by H with an extra
condition on the holonomy. These quotient spaces are orbifolds and the orbifold fundamental
group is a functor taking a morphism between orbifolds to a homomorphism between their
orbifold fundamental groups. Here, we reverse the process: given a group homomorphism,
is it coming from an orbifold morphism, immersion, or embedding from a 2-orbifold to
a 3-orbifold? We develop algorithms to decide this using orbifold handle decompositions,
extending normal surface theory, and applying the mapping class group.
## Representations of 3-Manifold GroupsExamples of Neumann's conjecture on the Bloch group## Research Report for Masters in Computer ScienceGeoRouteI am not responsible for the content of other pages that this page links to, nor do such pages necessarily agree with my own opinion. |