Zeta functions that hear the shape of a Riemann surface

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.

19 pages, 3 figures

References in corpus (2)

• Spectral triples for AF C*-algebras and metrics on the Cantor set
• Reconstruction of manifolds in noncommutative geometry

Cited by in corpus (3)

• Quantum Statistical Mechanics, L-series and Anabelian Geometry
• Modular Index Invariants of Mumford Curves
• Graphs, spectral triples and Dirac zeta functions