Bulletin of the American Mathematical Society

Author(s): Paul Schmutz Schaller
Journal: Bull. Amer. Math. Soc. 35 (1998), 193-214.
MSC (1991): Primary 30F45, 53C22, 57M50, 11F06, 11H99; Secondary 32G15, 11F72
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Abstract: The paper presents a survey on recent results on the geometry of Riemann surfaces showing that the study of closed geodesics provides a link between different aspects of Riemann surface theory such as hyperbolic geometry, topology, spectral theory, and the theory of arithmetic Fuchsian groups. Of particular interest are the systoles, the shortest closed geodesics of a surface; their study leads to the hyperbolic geometry of numbers with close analogues to classical sphere packing problems.

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