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Closed hyperbolic 3-manifolds whose closed geodesics all are simple. (English) Zbl 0783.53028

Let \(M\) be a complete orientable hyperbolic \(n\)-manifold of finite volume. A closed geodesic in \(M\) is called simple if it has no self- intersections. Otherwise a closed geodesic is called nonsimple. In the case of \(n=2\) any hyperbolic 2-manifold has nonsimple closed geodesics, indeed most closed geodesics in the case \(n = 2\) are nonsimple. The question has been asked in the case of \(n = 3\) as to whether there exist finite volume hyperbolic 3-manifolds all of whose closed geodesics are simple. The main result of this paper is to give a construction of infinitely examples of such manifolds. The examples are all closed. The construction is arithmetic in nature and uses the theory of quaternion algebras to interpret the existence of a nonsimple closed geodesic in terms of the Hilbert symbol of a particular quaternion algebra associated to a finite volume hyperbolic 3-manifold.
Reviewer: A.W.Reid (Austin)

MSC:

53C22 Geodesics in global differential geometry
53C20 Global Riemannian geometry, including pinching
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