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arXiv · 0708.0500

Zeta functions that hear the shape of a Riemann surface

Gunther Cornelissen·Matilde Marcolli
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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.

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Gunther Cornelissen, Matilde Marcolli. 2007-08-03. Zeta functions that hear the shape of a Riemann surface. https://doi.org/10.1016/j.geomphys.2007.12.011

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