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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.

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