Rigidity of length-angle spectrum for closed hyperbolic surfaces

TL;DR

This paper introduces the length-angle spectrum for closed hyperbolic surfaces and proves that it uniquely determines the surface, extending rigidity results beyond the marked length spectrum.

Contribution

It establishes the rigidity of the unmarked length-angle spectrum, a new invariant, for uniquely identifying closed hyperbolic surfaces.

Findings

Length-angle spectrum determines the surface uniquely.

Extends rigidity results to unmarked spectra.

Provides new tools for surface classification.

Abstract

The rigidity of marked length spectrum for closed hyperbolic surfaces due to Fricke-Klein [7] has been the motivation of many different rigidity results, specially for manifolds of negative curvature. From the works of Vigneras [18], Sunada [17] and many other authors this result is far from being true for the unmarked length spectrum. The purpose of this paper is to introduce a closely a related unmarked spectrum, the length-angle spectrum, and show that it determines the surface uniquely.