Abstract:In this paper we study kleinian groups of Schottky type whose limit set is a wild knot in the sense of Artin and Fox. We show that, if the "original knot" fibers over the circle then the wild knot Λ also fibers over the circle. As a consequence, the universal covering of S 3 − Λ is R 3 . We prove that the complement of a dynamically-defined fibered wild knot can not be a complete hyperbolic 3-manifold.
“…In this case one can show that the sphere is necessarily fractal (possibly unknotted). Examples of wild knots in S 3 , which are limit sets of geometrically finite Kleinian groups, were obtained by Maskit [22], Kapovich [17], Hinojosa [13], and Gromov, Lawson and Thurston [9]. An example of a wild 2-sphere in S 4 which is the limit set of a geometrically finite Kleinian group was obtained by the second-named author [14] and, independently, by Belegradek [5] (see also [4] for a wild limit set S 2 → S 3 ).…”
Abstract. In this paper we construct infinitely many wild knots, S n → S n+2 , for n = 1, 2, 3, 4 and 5, each of which is a limit set of a geometrically finite Kleinian group. We also describe some of their properties.
“…In this case one can show that the sphere is necessarily fractal (possibly unknotted). Examples of wild knots in S 3 , which are limit sets of geometrically finite Kleinian groups, were obtained by Maskit [22], Kapovich [17], Hinojosa [13], and Gromov, Lawson and Thurston [9]. An example of a wild 2-sphere in S 4 which is the limit set of a geometrically finite Kleinian group was obtained by the second-named author [14] and, independently, by Belegradek [5] (see also [4] for a wild limit set S 2 → S 3 ).…”
Abstract. In this paper we construct infinitely many wild knots, S n → S n+2 , for n = 1, 2, 3, 4 and 5, each of which is a limit set of a geometrically finite Kleinian group. We also describe some of their properties.
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