### Las Geometrías no euclídeas: Gauss, Lobatchewsky y Bolyai.

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In this paper, a representation of closed 3-manifolds as branched coverings of the 3-sphere, proved in [13], and showing a relationship between open 3-manifolds and wild knots and arcs will be illustrated by examples. It will be shown that there exist a 3-fold simple covering p : S --> S branched over the remarkable simple closed curve of Fox [4] (a wild knot). Moves are defined such that when applied to a branching set, the corresponding covering manifold remains unchanged, while the branching...

There is a disk in S whose interior is PL embedded and whose boundary has a tame Cantor set of locally wild points, such that the n-fold cyclic coverings of S branched over the boundary of the disk are all S. An uncountable set of inequivalent wild knots with these properties is exhibited.

It is proved that the Freudenthal compactification of an open, connected, oriented 3-manifold is a 3-fold branched covering of S, and in some cases, a 2-fold branched covering of S. The branching set is a locally finite disjoint union of strings.

This is a survey of some consequences of the fact that the fundamental group of the orbifold with singular set the Borromean link and isotropy cyclic of order 4 is a universal kleinian group.

A link L in S is universal if every closed, orientable 3-manifold is a covering of S branched over L. Thurston [1] proved that universal links exist and he asked if there is a universal knot, and also if the Whitehead link and the Figure-eight knot are universal. In [2], [3] we answered the first question by constructing a universal knot. The purpose of this paper is to prove that the Whitehead link and the Borromean rings, among others, are universal. The question about the Figure-eight knot remains...

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