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Geometrically finite hyperbolic structures on manifolds. (English) Zbl 0531.57012
The main aim of the present paper is to describe explicitly the properties of hyperbolic manifolds (perhaps with infinite volume) having geometrically finite structures. Let \(M^ n\), \(n\geq 2\), be a complete hyperbolic manifold and the holonomy be \(H: \pi_ 1(M^ n)\to G\subset Isom H^ n\). The structure of \(M^ n\) is called geometrical finite iff the discrete group G has a fundamental polyhedron in \(H^ n\) which is bounded by a finite number of faces. Denote as \(N\subset M^ n\) the minimal convex retract; \(N_{[\epsilon,\infty)}\) consists of points \(x\in N\) not lying on loops (nontrivial in N) of length \(<\epsilon.\)
Theorem. For some (or all) \(\epsilon>0\) the following statements are equivalent: 1. \(M^ n\) has the geometrical finite structure. 2. The \(\epsilon\)-neighbourhood \(U_{\epsilon}(N)\subset M^ n\) has finite volume. 3. The manifold \(N_{[\epsilon,\infty)}\) is compact. 4. The limit set L(G) of the group \(G=H(\pi_ 1(M^ n))\) consists of cusp points and conic approximation points.
Separate statements of this kind for \(n=3\) have been proved by A. Marsden, A. Beardon and B. Maskit, L. Greenberg, W. Thurston. The basic idea that allowed the author to prove these facts for \(n\geq 4\) is the investigation of the group’s action near parabolic vertices \((=cusp\) points) - introduced by the author [see Proc. Int. Conf. Complex Anal., Varna 1981, p. 88]. This action has for \(n\geq 4\) the interesting properties [see the author, ”Cusp ends of hyperbolic manifolds”, Ann. Global Anal. Geom. (to appear)]: there exist Kleinian groups G in \(\bar R^ n\), \(n\geq 3\), whose parabolic points p have no strictly \(G_ p\)- invariant horoballs \(B_ p\subset H^{n+1}\).

MSC:
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
53C30 Differential geometry of homogeneous manifolds
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
51M10 Hyperbolic and elliptic geometries (general) and generalizations
22E40 Discrete subgroups of Lie groups
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