On finite marked length spectral rigidity of hyperbolic cone surfaces and the Thurston metric

Abstract: We study the geometry of hyperbolic cone surfaces, possibly with cusps or geodesic boundaries. We prove that any hyperbolic cone structure on a surface of non-exceptional type is determined up to isotopy by the geodesic lengths of a finite specific homotopy classes of non-peripheral simple closed curves. As an application, we show that the Thurston asymmetric metric is well-defined on the Teichm\"uller space of hyperbolic cone surfaces with fixed cone angles and boundary lengths. We compare such a Teichm\"uller space with the Teichm\"uller space of complete hyperbolic surfaces with punctures, by showing that the two spaces (endowed with the Thurston metric) are almost isometric.