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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Multiplicities of simple closed geodesics and hypersurfaces in Teichmüller space

Greg McShane and Hugo Parlier

Geometry & Topology 12 (2008) 1883–1919
Abstract

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.

Keywords
simple closed geodesic, Teichmüller spaces, hyperbolic surface
Mathematical Subject Classification 2000
Primary: 57M50
Secondary: 58D99
References
Publication
Received: 25 July 2007
Revised: 1 May 2008
Accepted: 7 January 2008
Published: 5 July 2008
Proposed: Benson Farb
Seconded: Dave Gabai, Jean-Pierre Otal
Authors
Greg McShane
Laboratoire Emile Picard
Université Paul Sabatier
Toulouse
France
Hugo Parlier
Section de Mathématiques
École Polytechnique Fédérale de Lausanne
SB-IGAT-CTG, BCH
CH-1015 Lausanne
Switzerland