Harmonic manifolds and embedded surfaces arising from a super regular tesselation

Abstract

The main result of this paper is the construction of two Hyperbolic manifolds, M-1 and M-2, with several remarkable properties: (1) Every closed orientable 3-manifold is homeomorphic to the quotient space of the action of a group of order 16 on some covering space of M-1 or M-2. (2) M1 and M2 are tesselated by 16 dodecahedra such that the pentagonal faces of the dodecahedra fit together in a certain way. (3) There are 12 closed non-orientable hyperbolic surfaces of Euler characteristic -2 each of which is tesselated by regular right angled pentagons and embedded in M1 or M2. The union of the pentagonal faces of the tesselating dodecahedra equals the union of the 12 images of the embedded surfaces of Euler characteristic -2.