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Adiabatic limit of the eta invariant over cofinite quotients of PSL(2, ℝ)

Part of: Partial differential equations on manifolds; differential operators Lie groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  26 September 2008

Paul Loya ,
Sergiu Moroianu  and
Jinsung Park
Paul Loya
Affiliation:
Department of Mathematics, Binghamton University, Binghamton, NY 13902, USA (email: [email protected])
Sergiu Moroianu
Affiliation:
Institutul de Matematică al Academiei Române, PO Box 1–764, RO-014700 Bucharest, Romania (email: [email protected])
Jinsung Park
Affiliation:
School of Mathematics, Korea Institute for Advanced Study, 207–43, Cheongnyangni 2-dong, Dongdaemun-gu, Seoul 130–722, Korea (email: [email protected])
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Abstract

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The eta invariant of the Dirac operator over a non-compact cofinite quotient of PSL(2,ℝ) is defined through a regularized trace following Melrose. It reduces to the standard definition in terms of eigenvalues in the case of a totally non-trivial spin structure. When the S1-fibers are rescaled, the metric becomes of non-exact fibered-cusp type near the ends. We completely describe the continuous spectrum of the Dirac operator with respect to the rescaled metric and its dependence on the spin structure, and show that the adiabatic limit of the eta invariant is essentially the volume of the base hyperbolic Riemann surface with cusps, extending some of the results of Seade and Steer.

Keywords

eta invariantadiabatic limitfibered cusp operatorSelberg trace formula

MSC classification

Secondary: 58J28: Eta-invariants, Chern-Simons invariants 58J50: Spectral problems; spectral geometry; scattering theory 11F72: Spectral theory; Selberg trace formula 22E46: Semisimple Lie groups and their representations
Type
Research Article
Information
Compositio Mathematica , Volume 144 , Issue 6 , November 2008 , pp. 1593 - 1616
Copyright
Copyright © Foundation Compositio Mathematica 2008