Abstract
A hyperbolic 3-manifold is said to have the spd-property if all its closed geodesics are simple and pairwise disjoint. For a 3-manifold which supports a geometrically finite hyperbolic structure we show the following dichotomy: either the generic hyperbolic structure has the spd-property or no hyperbolic structure has the spd-property. Both cases are shown to occur. In particular, we prove that the generic hyperbolic structure on the interior of a handlebody (or a surface cross an interval) of negative Euler characteristic has the spd-property. Simplicity and disjointness are consequences of a variational result for hyperbolic surfaces. Namely, the intersection angle between closed geodesics varies nontrivially under deformation of a hyperbolic surface.
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Basmajian, A., Wolpert, S.A. Hyperbolic 3-Manifolds With Nonintersecting Closed Geodesics. Geometriae Dedicata 97, 251–257 (2003). https://doi.org/10.1023/A:1023657806258
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DOI: https://doi.org/10.1023/A:1023657806258