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Complexity of geodesics on 2-dimensional ideal polyhedra and isotopies

There are several ways of shortening a curve on a surface in order to establish the existence of closed geodesics. These methods are often useful in the study of much more general spaces, e.g. locally compact length spaces (see [1] chapter 10). Recently, J. Hass and P. Scott in [6] have introduced a new curve flow, the polygonal flow, which takes into consideration the complexity of the resulting geodesic. They construct the polygonal flow on surfaces so that the number of self-intersection points of a curve does not increase throughout the flow. In this paper we are concerned with ideal polyhedra with finitely many cells of dimension 2. These spaces consist of finite ideal hyperbolic triangles glued together by isometries along their sides.