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.
"Multiplicities of simple closed geodesics and hypersurfaces in Teichmüller space." Geom. Topol. 12 (4) 1883 - 1919, 2008. https://doi.org/10.2140/gt.2008.12.1883