Complexity of geodesics on 2-dimensional ideal polyhedra and isotopies.

*(English)*Zbl 0890.57047This paper deals with a question asked by V. Turaev on the number of double points within a homotopy between two curves having the same number of double points. For surfaces an answer was given by J. Hass and P. Scott in [Topology 33, No. 1, 25-43 (1994; Zbl 0798.58019)]. The result is that the number of double points does not necessarily increase during the homotopy, in other words: There is a homotopy with the same number of double points at each intermediate step. The authors prove a similar result for 2-dimensional ideal polyhedra which are built up by finitely many ideal hyperbolic triangles. They give an explicit formula for the number of additional double points. The method of a curve shortening flow introduced by Hass and Scott [loc. cit.] is decisively used.

Reviewer: W.Kühnel (Stuttgart)