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On the completion of incomplete hyperbolic structures. (Russian) Zbl 0744.51015
In a former conference report (1982), appeared by the author and V. S. Makarov [Mat. Issled. 103, 69-76 (1988; Zbl 0669.51015)], the authors deformed the complete hyperbolic octahedron space (on the Whitehead link complement), so that it becomes an incomplete manifold with locally hyperbolic structure, depending on one continuous parameter \(t\). (For the theory see the monograph of B. N. Apanasov, Discrete groups in space and uniformization problems, Dordrecht (1991; Zbl 0732.57001), Ch. 2 and 6).
In this paper the author gives an elegant synthetic geometric approach showing how to choose the parameter \(t\), so that the group generated by the face identifications of the deformed ideal octahedron will be a discrete group of hyperbolic motions. He determines also a fundamental polyhedron for this group that represents a noncompact hyperbolic orbifold of finite volume. It is not a manifold because of \(k\)-fold rotation with \(k\geq 3\) depending on \(t\) (not expressed explicitly in this paper).
51M20 Polyhedra and polytopes; regular figures, division of spaces
51M10 Hyperbolic and elliptic geometries (general) and generalizations
52B70 Polyhedral manifolds
57S30 Discontinuous groups of transformations
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