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Zeta functions that hear the shape of a Riemann surface

Cornelissen, Gunther and Marcolli, Matilde (2008) Zeta functions that hear the shape of a Riemann surface. Journal of Geometry and Physics, 58 (5). pp. 619-632. ISSN 0393-0440.

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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 measured. 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-)complex isomorphism class of the corresponding Riemann surface. Thus, you can hear the complex analytic shape of a Riemann surface, by listening to a suitable spectral triple.

Item Type:Article
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Additional Information:© 2008 Elsevier Ltd. Received 9 November 2007; revised 17 December 2007; accepted 30 December 2007. Available online 6 January 2008.
Subject Keywords:non-commutative geometry; spectral triples; Kleinian-groups; manifolds; algebras; rigidity; curves; set
Issue or Number:5
Classification Code:MSC: 20H10; 57S30; 58B34
Record Number:CaltechAUTHORS:CORjgp08
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:13544
Deposited By: Tony Diaz
Deposited On:08 May 2009 15:55
Last Modified:03 Oct 2019 00:40

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