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, [Formula: see text] and [Formula: see text], with several remarkable properties: (1) Every closed orientable [Formula: see text]-manifold is homeomorphic to the quotient space of the action of a group of order [Formula: see text] on some covering space of [Formula: see text] or [Formula: see text]. (2) [Formula: see text] and [Formula: see text] 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 [Formula: see text] each of which is tesselated by regular right angled pentagons and embedded in [Formula: see text] or [Formula: see text]. The union of the pentagonal faces of the tesselating dodecahedra equals the union of the 12 images of the embedded surfaces of Euler characteristic [Formula: see text].