Multiplicities of simple closed geodesics and hypersurfaces in Teichmüller space

McShane, Greg¹ ; Parlier, Hugo²

¹ Laboratoire Emile Picard, Université Paul Sabatier, Toulouse, France
² Section de Mathématiques, École Polytechnique Fédérale de Lausanne, SB-IGAT-CTG, BCH, CH-1015 Lausanne, Switzerland

Geometry & topology, Tome 12 (2008) no. 4, pp. 1883-1919.

Résumé

Using geodesic length functions, we define a natural family of real codimension 1 subvarieties of Teichmüller space, namely the subsets where the lengths of two distinct simple closed geodesics are of equal length. We investigate the point set topology of the union of all such hypersurfaces using elementary methods. Finally, this analysis is applied to investigate the nature of the Markoff conjecture.

Détail

DOI : 10.2140/gt.2008.12.1883

Keywords: simple closed geodesic, Teichmüller spaces, hyperbolic surface

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