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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).
##### MSC:
 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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