**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

E-mail: geometry@ccmail.bme.hu

**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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