Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 39, No. 2, pp. 497--516 (1998)

Classification of dodecahedral space forms

István Prok

TU Budapest, Department of Geometry
H--1521 Budapest XI, Egry J. u. 1, H.II.22, Hungary

Abstract: Considering the regular dodecahedron tilings in the spaces of constant curvature we look for the fixed point free discrete transformation groups that act simply transitively on the tiles. The isometries of a group, mapping a distinguished tile onto its neighbours, identify the faces and determine a dodecahedral space form, moreover, these transformations generate this fundamental group. In order to find all these groups, in this paper we apply a polyhedron algorithm, which was developed theoretically by E. Molnár and implemented to computer by the author. Because of the large number of cases we modify the general algorithm and apply it for finding all the non-equivariant face identifications of a regular polyhedron, and for choosing those ones that generate fixed point free groups, if the number of polyhedra around the edges is given. In this way we obtain the fundamental groups of dodeca\-hedral space forms given by generators and relations. We classify these non-equivariant forms according to the obvious orientability and to the first homology groups. When the vertices are ideal points we shall take into consideration the so-called cusp structure refining our classification. The fundamental groups in different classes will be non-isomorphic, but we shall leave the general problem of isomorphism within the classes open for later papers. The isomorphism problem has already been solved for hyperbolic octahedron spaces recently. So this paper gives a complete list of regular dodecahedral space forms, but in some small classes the difference of the manifolds (up to isometry) cannot be guaranteed yet. From the results we emphasize that we improved the list of compact dodecahedron spaces by L. A. Best, and we extended the investigations for non-orientable space forms, too.

Keywords: (regular) dodecahedron; regular tilings; (classification) hyperbolic space; spherical space; polyhedral manifolds

Classification (MSC91): 52C22; 52B70; 51M20; 52B10; 57M50

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