Some curious Kleinian groups and hyperbolic 5-manifolds
The so-called ``suspension technique over embedded necklaces'' [\textit{N. H. Kuiper}, `Geometry and topology of submanifolds II', Proc. Int. Meet., Avignon/Fr. 1988, 165-204 (1990; Zbl 0731.57007)] is used to produce an interesting family of discrete subgroups of SO(5,1) isomorphic to the lattices of SO(3,1): their action on \(\mathbb{H}^5\) allows to construct complete hyperbolic 5-manifolds that are nontrivial plane bundles over closed hyperbolic 3-manifolds and conformally flat 4-manifolds that are nontrivial circle bundles over closed hyperbolic 3-manifolds. Moreover, suitable choices for embedded necklaces yield, as limit sets of the associated groups, either unknotted 2-spheres or wildly knotted 2-spheres in \(\partial_\infty \mathbb{H}^5\) (see [\textit{M. Gromov, H. B. Lawson jun.} and \textit{W. Thurston}, Publ. Math., Inst. Hautes Étud Sci. 68, 27-45 (1988; Zbl 0692.57012); \textit{B. N. Apanasov} and \textit{A. V. Tetenov}, J. Differ. Geom. 28, No. 3, 407-422 (1988; Zbl 0627.57015)] for analogous results in dimension four). As a consequence, a closed hyperbolic 3-manifold \(M\) is proved to exist, such that the space \(CC^5(\pi_1(M))\) of convex cocompact representations of its fundamental group is not connected.
scientific article; zbMATH DE number 1139551