Geodesic
Minimal geodesics of a torus with a hole
Izvestiya. Mathematics, Tome 29 (1987) no. 2, pp. 449-457 
H. Zieschang. Minimal geodesics of a torus with a hole. Izvestiya. Mathematics, Tome 29 (1987) no. 2, pp. 449-457. http://geodesic.mathdoc.fr/item/IM2_1987_29_2_a8/
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     title = {Minimal geodesics of a~torus with a~hole},
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $R$ be a Riemann surface of genus $1$ with one hole. For a given homology class $\alpha\in H_1(R)$, the author determines that homotopy class within $\alpha$ which contains the shortest curve in $\alpha$. It turns out that this homotopy class is uniquely determined independently of the Riemannian metric. A conjecture of H. Cohn is thereby confirmed. Bibliography: 7 titles.

[1] Cohn H., “Markoff forms and primitive words”, Math. Ann., 196 (1972), 8–22 | DOI | MR | Zbl

[2] Cohn H., “Minimal geodesics on Fricke's torus-covering”, Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978,), Ann. of Math. Stud., 97, Princeton Univ. Press, Princeton, N.J., 1981, 73–85 | MR

[3] Horowitz R. D., “Characters of free groups represented in the two dimensional special linear group”, Commun. Pure Appl. Math., 25 (1972), 635–649 | DOI | MR

[4] Osborne R. P., Zieschang H., “Primitives in the free group on two generators”, Invent. math., 63 (1981), 17–24 | DOI | MR | Zbl

[5] Zieschang H., Finite Groups of Mapping Classes, Lecture Notes in Math., 875, Springer-Verlag, Berlin, Heidelberg, New York, 1981 | MR | Zbl

[6] Zieschang H., Vogt E., Coldewey H.-D., Surfaces and Planar Discontinuous Groups, Lecture Notes in Math., 835, Springer-Verlag, Berlin, Heidelberg, New York, 1980 | MR | Zbl

[7] Haas A., Length spectra as moduli for hyperbolic surfaces, Preprint | MR