UC Berkeley

We study embeddings of regular tessellations into S^S such that some symmetries of the tessellation are directly visible in space. In the first chapter, we consider cusped hyperbolic 3-manifolds which arise from principal congruence subgroups and, therefore, are canonically tessellated by regular ideal hyperbolic tetrahedra. The codimension of an embedding of such a 3-manifold into S^S is zero, and the embedding fills all of S^S but a link, i.e., a disjoint union of knots. A new example constructed here is the 12-component link whose complement consists of 54 regular ideal hyperbolic tetrahedra. The link has 3-fold dihedral symmetry making some of the symmetries of its hyperbolic complement directly visible in the picture. In the second chapter, we embed surfaces F into 3-space. These surfaces are regular maps, i.e., regular tessellations by polygons.

The main result of the first chapter is the construction of two new principal congruence links. These links have hyperbolic complements arising from a natural number theoretic construction to obtain regular covers of Bianchi orbifolds. The datum for the construction of these arithmetic 3-manifolds is an ideal in the ring of integers O_D in the imaginary quadratic number field Q(SQRT(D)) of discriminant D<0. Results by Vogtman, Lackenby, and Agol imply that there are only finitely many principal congruence links. Here, we attempt to list all of the principal congruence links for the discriminant D=-3. Previously, only two such principal congruence links due to Dunfield and Thurston were known for the prime ideals generated by z=2 and z=2+u. We show that there are at most seven principal congruence links and explicitly construct two more for the non-prime ideals z=2+2u and z=3. For the construction, we use that the deck transformation group of these Bianchi orbifold covers is a solvable extension to break down the construction into a sequence of branched cyclic covers starting from a known principal congruence orbifold diagram. Each cyclic cover can be obtained by either using Akbulut and Kirby's construction or by unfolding the Euclidean (3,3,3)-triangle orbifold. The chapter finishes with a discussion of generic regular covers of the Bianchi orbifold for O_-3 by explicitly constructing the category of all such covers with small fixed cusp modulus.

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 defined 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 definition, 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 fulfill 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.

Further connections intimately tie the two chapters together: For example, restricting the canonical triangulation of a principal congruence manifold to a cusp induces a regular map on the cusp torus, yielding an invariant we call cusp modulus. Furthermore, the 2-skeleton of the canonical triangulation of a principal congruence manifold is an immersed regular map, e.g., the Klein quartic in case of the complement of the Thurston congruence link.