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Closed geodesics on ideal polyhedra of dimension 2. (English) Zbl 0869.52003
Ideal polyhedra of dimension 2 are unions of ideal hyperbolic triangles which are glued together by isometries along their sides. The infimum of the lengths of broken geodesics defines a pseudodistance on such ideal polyhedra.
In Theorem 1 the author proves that this is in fact a geodesic metric under certain finiteness assumptions. In Theorem 2 it is shown that for each closed curve there is a unique distance realizing closed geodesic in the same homotopy class. The only exception are curves homotopic to a cusp, meaning that the infimum of the length in this homotopy class is zero. However, this minimal closed geodesic is not necessarily simple.

52A55 Spherical and hyperbolic convexity
53C22 Geodesics in global differential geometry
51M10 Hyperbolic and elliptic geometries (general) and generalizations
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