Filling a space by polyhedra and deformation of incomplete hyperbolic structures
The author obtains conditions for complete filling of hyperbolic spaces by unbounded polyhedra with finite numbers of sides for dimensions \(n\leq 3\), and finite volumes for dimensions \(n\geq 4\). In dimensions \(n\leq 3\) these conditions are the conditions in A. D. Aleksandrov's theorem for complete filling of hyperbolic spaces by bounded polyhedra, together with the local finiteness condition for the filling. In dimensions \(n\geq 4\) these conditions for unbounded polyhedra topologically similar to polyhedra with a complete filling are the same as the conditions in the Aleksandrov theorem. These results could be considered as generalizations of the classical Poincaré theorems about fundamental polygons and polyhedra. The methods used in this paper are closely connected to the study of the deformations of complete and incomplete hyperbolic manifolds and orbifolds. A strong rigidity for such manifolds in dimensions \(n\geq 4\) is obtained.
filling of hyperbolic spaces by unbounded polyhedra
deformations of complete and incomplete hyperbolic manifolds and orbifolds
scientific article; zbMATH DE number 4092372
scientific article