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

## TL;DR

This paper proves that hyperbolic cone surfaces are uniquely determined by finitely many geodesic lengths and demonstrates that the Thurston metric extends to their Teichmüller space, nearly isometrically relating it to punctured hyperbolic surfaces.

## Contribution

It establishes finite spectral rigidity for hyperbolic cone surfaces and extends the Thurston metric to their Teichmüller space, comparing it with classical hyperbolic surfaces.

## Key findings

- Hyperbolic cone structures are determined by finitely many geodesic lengths.
- The Thurston metric is well-defined on the Teichmüller space of cone surfaces.
- The Teichmüller space of cone surfaces is almost isometric to that of punctured hyperbolic surfaces.

## 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.