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Parabolic vertices and the property of finiteness for Kleinian groups in space. (Russian) Zbl 0567.57008
The paper gives a simplified proof of geometric finiteness criteria of the hyperbolic structure of a manifold $$M^ n$$, $$n\geq 4$$ [they were obtained earlier by the author, Sib. Mat. Zh. 23, No.6, 16-27 (1982; Zbl 0519.30038); Ann. Global Anal. Geom. 1, No.3, 1-22 (1983; Zbl 0531.57012), see also his book ”Discrete transformation groups and manifold structures”, Nauka, Novosibirsk (1983)]. The principal distinction of this case from the known one before, $$n\leq 3$$ [A. Beardon and B. Maskit, Acta Math. 132, 1-12 (1974; Zbl 0277.30017); A. Marden, Ann. Math., II. Ser. 99, 383-462 (1974; Zbl 0282.30014); W. P. Thurston ”The geometry and topology of 3-manifolds”, Mimeographed Math. Notes, Princeton, Chapter 8.4], is due to the fact that at $$n\geq 4$$ the ”thin” ends of the manifold $$H^ n/G$$ are not, generally speaking, the horoball factor by the action of the Bieberbach group [see the author’s paper Ann. Global Anal. Geom. 3, 1-11 (1985; Zbl 0537.57007)].
The methods of the paper and the results of D. Sullivan [Publ. Math., Inst. Hautes Étud. Sci. 50, 171-202 (1979; Zbl 0439.30034)] also allow the author to obtain the rigidity condition of deformations of n- dimensional hyperbolic manifolds with boundary (Theorem 9.9). The exactness of these conditions is obtained by modification of the author’s wellknown construction [Dokl. Akad. Nauk SSSR 243, 829-832 (1978; Zbl 0414.20038)].

##### MSC:
 57N15 Topology of the Euclidean $$n$$-space, $$n$$-manifolds ($$4 \leq n \leq \infty$$) (MSC2010) 57S30 Discontinuous groups of transformations 51M10 Hyperbolic and elliptic geometries (general) and generalizations 22E40 Discrete subgroups of Lie groups
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