Every closed flat 3-manifold is realized as a cusp section of a cusp-transitive finite-volume hyperbolic 4-manifold; dense subsets of flat metrics on each such 3-manifold are also realizable, and many 4-manifolds exist with pairwise isometric cusps of any given type.
Describing the platycosms
2 Pith papers cite this work. Polarity classification is still indexing.
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abstract
We study in detail the closed flat Riemannian 3-manifolds.
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Holonomy groups of compact flat solvmanifolds are exactly the finite abelian groups, with minimal dimensions for cyclic groups matching those of general compact flat manifolds.
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Cusp-transitive 4-manifolds with every cusp section
Every closed flat 3-manifold is realized as a cusp section of a cusp-transitive finite-volume hyperbolic 4-manifold; dense subsets of flat metrics on each such 3-manifold are also realizable, and many 4-manifolds exist with pairwise isometric cusps of any given type.
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Holonomy groups of compact flat solvmanifolds
Holonomy groups of compact flat solvmanifolds are exactly the finite abelian groups, with minimal dimensions for cyclic groups matching those of general compact flat manifolds.