Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 42, No. 1, pp. 235-250 (2001)

The Classification of $ S^2\times R$ Space Groups

J. Z. Farkas

Budapest University of Technology and Economics, Institute of Mathematics, Department of Geometry, Budapest XI. Egry J. str. 1 H1521, Hungary, e-mail:

Abstract: The geometrization of 3-manifolds plays an important role in various topological investigations and in the geometry as well. Thurston classified the eight simply connected 3-dimensional maximal homogeneous Riemannian geometries [A], [B]. One of these is $ S^2\times R$, i.e. the direct product of the spherical plane $ S^2$ and the real line $\bf R$. Our purpose is the classification of the space groups of $ S^2\times R$, i.e. discrete transformation groups which act on $ S^2\times R$ with a lattice on $\bf R$ (see Section 3), analogously to that of the classical Euclidean geometry $ E^3$.

\item{[A]} Thurston, W. P.: Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. 6 (1982), 357-381.

\item{[B]} Thurston, W.P.: Three-dimensional geometry and topology. Vol.1, ed. by S. Levy, Princeton University Press 1997, (Ch.3.8,4.7).

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