Zeta functions that hear the shape of a Riemann surface

TL;DR

This paper constructs a spectral triple associated with a hyperbolic Riemann surface and shows that its zeta functions uniquely determine the surface's conformal class, effectively allowing one to 'hear' the shape of the surface.

Contribution

It introduces a spectral triple framework for hyperbolic Riemann surfaces and proves that zeta functions encode the surface's conformal geometry, linking spectral data to geometric classification.

Findings

Spectral triples encode boundary actions of Schottky groups.

Zeta functions characterize the conformal class of Riemann surfaces.

Ergodic rigidity implies spectral data suffices for surface identification.

Abstract

To a compact hyperbolic Riemann surface, we associate a finitely summable spectral triple whose underlying topological space is the limit set of a corresponding Schottky group, and whose ``Riemannian'' aspect (Hilbert space and Dirac operator) encode the boundary action through its Patterson-Sullivan measure. We prove that the ergodic rigidity theorem for this boundary action implies that the zeta functions of the spectral triple suffice to characterize the (anti-)conformal isomorphism class of the corresponding Riemann surface. Thus, you can hear the shape of a Riemann surface, by listening to a suitable spectral triple.