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Equivalent curves in surfaces. (English) Zbl 1037.57013
Given a closed orientable surface S of genus greater than 1 and the curves \(\gamma\) and \(\gamma^{\prime}\) on \(S\), the author discusses the following two problems: 1. Find a topological description of pairs of distinct homotopy classes of curves \(\gamma\) and \(\gamma^{\prime}\) which have the same length with respect to every hyperbolic structure on S. 2. Do there exist pairs of distinct homotopy classes of curves \(\gamma\) and \(\gamma^{\prime}\) which have the same length with respect to every metric in a family of path metrics on \(S\)? If so, find a topological description of such pairs.
The author considers various notions of equivalence for homotopy classes of curves on hyperbolic surfaces based on topological, algebraic and geometric structures and finds the relationships between these equivalences. The second problem is discussed in the context of both, hyperbolic and branched flat metrics, and a complete solution is given in the latter case. He also shows that the ”obvious” necessary topological condition with respect to the hyperbolic metrics is not sufficient, gives a possible alternative and constructs counter-examples completing the proof of the main result. The author also presents a few difficulties encountered in finding such a pair of curves and describes a candidate for a topological characterization of hyperbolic equivalence. He ends this paper by citing references for more information on the first problem and other related problems.

MSC:
57M99 General low-dimensional topology
57M50 General geometric structures on low-dimensional manifolds
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