Cyclic coverings of the 3-sphere branched over wild knots of dynamically defined type
The authors study cyclic branched coverings of the 3-sphere whose branched sets are ``wild knots of dynamically defined type'', i.e. wild knots obtained via the action of Kleinian groups on the 3-sphere, and prove some topological properties of such knots. ``We will construct, via the action of a Kleinian group, a sequence of nested pearl chain necklaces whose inverse limit is a wild knot of dynamically defined type; in particular, we generalize the construction of cyclic branched coverings for this case, and we show that there exists a wild knot of dynamically defined type such that \(S^3\) is an \(n\)-fold cyclic branched covering of \(S^3\) along it, for \(n \ge 2\).'' \N\NIn such a context, \textit{J. M. Montesinos-Amilibia} [Rev. Mat. Complut. 16, No. 1, 329--344 (2003; Zbl 1052.57015)] proved that there are uncountably many wild knots whose cyclic branched coverings are \(S^3\), and in particular uncountably many periodic homeomorphisms of \(S^3\) whose fixed point sets are wild knots (cf. the solution of the classical Smith conjecture for periodic diffeomorphisms of \(S^3\)).
scientific article; zbMATH DE number 7878882
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