Classification of dodecahedral space forms (Q1267349)

This paper is concerned with regular tilings of spherical or hyperbolic 3-space, by (congruent regular) dodecahedra, and their classification by systematic combinatorial search. The author describes the theoretical background, the algorithm, and presents the results obtained by a computer-aided enumeration which is based on the number \(n=3,5,6\) of dodecahedra along each edge and a characteristic pairing of the twelve faces of a dodecahedron, induced by groups of (fixed point free) symmetries of the tiling. There are 147~essentially different (\textit{non-equivariant}) pairings (for each of which a million cases has to be checked). For \(n=3\) two distinct orientable spherical manifolds are found; for \(n=5\) the enumeration yields 12~compact orientable hyperbolic manifolds, of which at least 7 are distinct because they have distinct first homology groups; and for \(n=6\) there are 78~non-compact hyperbolic manifolds (10~orientable and 68~non-orientable), of which at least~38 (7~and 31, respectively) are distinct (due to distinct homology groups or distinct cusp structures). The details of these results are presented in several tables. They confirm and extend previous enumerations. A typical sample entry is illustrated and discussed in detail. This paper can be viewed as part of a series of papers (on related topics) by the author and Emil Molnár.