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 H3 divided by a principal congruence subgroup of PGL(2,OD) where OD is the ring of integers of the imaginary quadratic number field of discrimninant D=-3. The groups PGL(2,OD), respectively, PSL(2,OD) (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,OD). They are given by ker(PGL(2,OD) ->PGL(2,OD/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 E3. 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 E3. 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 E3 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 E3. 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 E3 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 Science