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

Published online by Cambridge University Press:  26 September 2008


Paul Loya
Affiliation:
Department of Mathematics, Binghamton University, Binghamton, NY 13902, USA (email: paul@math.binghamton.edu)
Sergiu Moroianu
Affiliation:
Institutul de Matematică al Academiei Române, PO Box 1–764, RO-014700 Bucharest, Romania (email: moroianu@alum.mit.edu)
Jinsung Park
Affiliation:
School of Mathematics, Korea Institute for Advanced Study, 207–43, Cheongnyangni 2-dong, Dongdaemun-gu, Seoul 130–722, Korea (email: jinsung@kias.re.kr)

Abstract

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.


Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2008

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